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                     P-BLTZMC08_805-872-hr  21-11-2008  13:26  Page 856
                                 856 Chapter 8 Matrices and Determinants
                                                    35 -1-1                                                                        Preview Exercises
                                                                      1A
                                 90. If find A =             ,              2 .
                                                  B        R
                                                    24                                                                             Exercises 93–95 will help you prepare for the material covered in
                                 91. Find values of afor which the following matrix is not invertible:                             the next section.Simplify the expression in each exercise.
                                                                     B        1    a + 1R.                                         93. 21-52 - 1-32142
                                                                       a - 24                                                            21-52 - 11-42
                                                                                                                                   94. 51-52 - 61-42
                                 Group Exercise                                                                                    95. 21-30 - 1-322 - 316 - 92 + 1-1211 - 152
                                 92. Each person in the group should work with one partner.
                                       Send a coded word or message to each other by giving your
                                       partner the coded matrix and the coding matrix that you
                                       selected.Once messages are sent,each person should decode
                                       the message received.
                                                     Section
                                                                             8.5 Determinants and Cramer’s Rule
                                       Objectives                                                                                                                       A portion of Charles Babbage’s unrealized
                                    Evaluate a second-order                                                                                                            Difference Engine
                                           determinant.
                                    Solve a system of linear
                                           equations in two variables                                                                                                   s cyberspace absorbs more and
                                           using Cramer’s rule.                                                                                                  Amore of our work, play, shopping,
                                    Evaluate a third-order                                                                                                    and socializing, where will it all end?
                                           determinant.                                                                                                      Which activities will still be offline in
                                    Solve a system of linear                                                                                                2025?
                                           equations in three variables                                                                                                   Our technologically transformed
                                           using Cramer’s rule.                                                                                                   lives can be traced back to the English
                                    Use determinants to identify                                                                                                 inventor Charles Babbage (1792–1871).
                                           inconsistent systems and                                                                                               Babbage knew of a method for
                                           systems with dependent                           solving linear systems called Cramer’s rule,in honor of the Swiss geometer Gabriel
                                           equations.                                       Cramer (1704–1752). Cramer’s rule was simple, but involved numerous
                                                                                            multiplications for large systems. Babbage designed a machine, called the
                                    Evaluate higher-order                                  “difference engine,” that consisted of toothed wheels on shafts for performing
                                           determinants.                                    these multiplications. Despite the fact that only one-seventh of the functions ever
                                                                                            worked, Babbage’s invention demonstrated how complex calculations could be
                                                                                            handled mechanically.In 1944,scientists at IBM used the lessons of the difference
                                                                                            engine to create the world’s first computer.
                                                                                                     Those who invented computers hoped to relegate the drudgery of repeated
                                                                                            computation to a machine. In this section, we look at a method for solving linear
                                                                                            systems that played a critical role in this process. The method uses real numbers,
                                                                                            called determinants, that are associated with arrays of numbers. As with matrix
                                                                                            methods,solutions are obtained by writing down the coefficients and constants of a
                                                                                            linear system and performing operations with them.
                                 Evaluate a second-order                                   The Determinant of a 2 : 2 Matrix
                                         determinant.
                                                                                            Associated with every square matrix is a real number, called its determinant.The
                                                                                            determinant for a 2 * 2 square matrix is defined as follows:
                                                                                                                  P-BLTZMC08_805-872-hr  21-11-2008  13:26  Page 857
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Section 8.5 Determinants and Cramer’s Rule                                                                                                                                                                                                                                                                                                                                                                                857
                                                                                                                                                                                                                        Study Tip                                                                                                                                                                                                                                                                                                                                         Definition of the Determinant of a 2 : 2 Matrix
                                                                                                                                                                                                                        To evaluate a second-order                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          a                                         b                                                                                                                                                                a                                         b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          The determinant of the matrix                                                                                                                                                                                                                                                                1                                        1                         is denoted by                                                                                                                           1                                        1 and is defined by
                                                                                                                                                                                                                        determinant,find the difference of the                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  Ba                                                    b R                                                                                                                                                         `    a                                         b `
                                                                                                                                                                                                                        product of the two diagonals.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  2                                        2                                                                                                                                                                 2                                        2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   a                                         b
                                                                                                                                                                                                                                                                                    a                                 b                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       1                                        1
                                                                                                                                                                                                                                                                                             1                                 1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              `                                                                   `             = a b - a b .
                                                                                                                                                                                                                                                                                                                                                                        a b                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    1 2                                                              2 1
                                                                                                                                                                                                                                                                               `                                                         `           =-ab
                                                                                                                                                                                                                                                                                                                                                                                 1 2                                                        2 1                                                                                                                                                                                                                                                                                                                                                                                    a                                         b
                                                                                                                                                                                                                                                                                    a                                 b                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       2                                        2
                                                                                                                                                                                                                                                                                             2                                 2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 a                                         b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          We also say that the value of the second-order determinant ` 1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             1 `                         is
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 a                                         b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            2                                        2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           a b - a b .
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     1 2                                                              2 1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Example 1 illustrates that the determinant of a matrix may be positive or
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            negative.A determinant can also have 0 as its value.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            EXAMPLE 1 Evaluating the Determinant of a 2 : 2 Matrix
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Evaluate the determinant of each of the following matrices:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     56                                                                                                                                                                                                                                             24
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          a.                                                                                                                                                                                                                                b.                                                                                                                                .
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         B                                                         R                                                                                                                                                                     B                                                                                        R
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     73                                                                                                                                                                                                                                -3                                               -5
                                                                                                                                                                                                                        Discovery                                                                                                                                                                                                                                                                                                                           Solution                                                                                                     We multiply and subtract as indicated.
                                                                                                                                                                                                                        Write and then evaluate three                                                                                                                                                                                                                                                                                                                                                                                         "                                                                                                                                                                                                                                                                                                                                                                                                                                 The value of the second-
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                56
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                        determinants, one whose value is                                                                                                                                                                                                                                                                                                                                                                                                                                                         #                                                               #
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          a.                               `             "                                      `             = 5 3 - 7 6 = 15 - 42 =-27                                                                                                                                                                                                                                                                                                                                                                        order determinant is -27.
                                                                                                                                                                                                                        positive,one whose value is negative,                                                                                                                                                                                                                                                                                                                                                   73
                                                                                                                                                                                                                        and one whose value is 0.                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          "
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                24                                                                                                                                                                                                                                                                                                                                                                                                                                              The value of the second-
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         b.                                `                                      "                                            `             = 21-52 - 1-32142 =-10 + 12 = 2                                                                                                                                                                                                                                                                                                                                    order determinant is 2.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  -3                                                -5
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             Check Point 1 Evaluate the determinant of each of the following matrices:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       10                                          9                                                                                                                                                                                                                            43
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            a.                                                                                                                                                                                                                                                          b.                                                                                                                                .
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           B                                                                   R                                                                                                                                                                                    B                                                                                         R
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 65                                                                                                                                                                                                                                               -5                                                -8
                                                                                                                                                                                                                         Solve a system of linear                                                                                                                                                                                                                                                                                                          Solving Systems of Linear Equations 
                                                                                                                                                                                                                                                                  equations in two variables                                                                                                                                                                                                                                                                in Two Variables Using Determinants
                                                                                                                                                                                                                                                                  using Cramer’s rule.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Determinants can be used to solve a linear system in two variables.In general,such
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            a system appears as
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      a x + b y = c
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 1                                                         1                                                       1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          ba x + b y = c .
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 2                                                         2                                                       2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Let’s first solve this system for x using the addition method.We can solve for x by
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            eliminating y from the equations. Multiply the first equation by b and the second
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            equation by -b .Then add the two equations:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          Multiply by b .  
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        a x + b y = c                                                                                                                                                                                                                                                                       2                                  "                                                                  a b x + b b y =                                                                                                                                                        c b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   1                                                         1                                                       1                                                                                                                                                                                                                                                                       1 2                                                                        1 2                                                                                      1 2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           b                                                                                                                                                                             Multiply by -b .                                                                                                                                                                                            -a b x - b b y =-c b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        a x + b y = c                                                                                                                                                                                                                                                                                 1                         "                                                                            2 1                                                                        1 2                                                                                      2 1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   2                                                         2                                                       2                                                                                                       
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Add:                                                                        1a b - a b 2x =                                                                                                                                                               c b - c b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            1 2                                                             2 1                                                                                                1 2                                                           2 1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         c b - c b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              x =                                                                 1 2                                                           2 1 .
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       a b - a b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  1 2                                                             2 1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Because
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           c                                       b                                                                                                                                                                                                                                       a                                         b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      `             1                                        1 `                      = c b - c b   and  ` 1                                                                                                                                                                                                                                   1 `                      = a b - a b ,
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           c                                       b                                                               1 2                                                            2 1                                                                                                      a                                         b                                                                 1 2                                                              2 1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    2                                        2                                                                                                                                                                                                                                        2                                        2
                                                                                                                  P-BLTZMC08_805-872-hr  21-11-2008  13:26  Page 858
                                                                                                                                                                                    858 Chapter 8 Matrices and Determinants
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         we can express our answer for x as the quotient of two determinants:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       c                                      b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  `            1                                        1 `
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            c b - c b                                                                                                                                  c                                      b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                x = 1 2                                                                                                            2 1 =                                                                       2                                        2 .
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          a b - a b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    1 2                                                              2 1                                                              a                                        b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 `              1                                        1 `
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      a                                        b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                2                                        2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         Similarly,we could use the addition method to solve our system for y, again expressing y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         as the quotient of two determinants. This method of using determinants to solve the
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         linear system,called Cramer’s rule, is summarized in the box.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      Solving a Linear System in Two Variables Using Determinants
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      Cramer’s Rule
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      If
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      a x + b y = c
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                1                                                          1                                                       1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       ba x + b y = c ,
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              2                                                         2                                                       2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      then
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 c                                       b                                                                                                                                                                                               a                                        c
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            `             1                                        1 `                                                                                                                                                                              `              1                                       1 `
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 c                                       b                                                                                                                                                                                               a                                        c
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             x =                                                          2                                        2  and y = 2 2 ,
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                a                                         b                                                                                                                                                                                             a                                         b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           `               1                                        1 `                                                                                                                                                                            `               1                                        1 `
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                a                                         b                                                                                                                                                                                             a                                         b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           2                                        2                                                                                                                                                                                              2                                        2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      where
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             a                                        b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        `              1                                        1 `                      Z 0.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             a                                        b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       2                                        2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         Here are some helpful tips when solving
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      a x + b y = c
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                1                                                          1                                                       1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         ba x + b y = c
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                2                                                          2                                                       2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         using determinants:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           1. Three different determinants are used to find x and y.The determinants in the
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  denominators for x and y are identical.The determinants in the numerators
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  for x and y differ.In abbreviated notation,we write
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  D
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        D                                                                                                                                                        y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              x =                                                      x  and  y =                                                                                                                                          , where D Z 0.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           D                                                                                                                                         D
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          2. The elements of D, the determinant in the denominator,are the coefficients of
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  the variables in the system.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               a                                         b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        D = ` 1                                                                                                    1 `
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               a                                         b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          2                                        2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          3. D , the determinant in the numerator of x, is obtained by replacing the
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   x-coefficients, in D, a1 and a2, with the constants on the right sides of the
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  equations,c and c .
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         1                                                          2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          a                                         b                                                                                                                                                               c                                      b                                                  Replace the column with a and a with 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   D = ` 1                                                                                                    1 `   and  D = ` 1                                                                                                                                                                                     1 `                                                                                                                                                                                                                                         1                                                  2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          a                                         b                                                                                                                x                                              c                                      b                                                  the constants c1and c2 to get Dx.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     2                                        2                                                                                                                                                             2                                        2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          4. D , the determinant in the numerator for y, is obtained by replacing the
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   y-coefficients, in D, b and b , with the constants on the right sides of the
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   1                                                                  2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  equations,c and c .
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         1                                                          2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          a                                         b                                                                                                                                                              a                                         c                                             Replace the column with b and b with 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   D = ` 1                                                                                                    1 `   and  D = ` 1                                                                                                                                                                                      1 `                                                                                                                                                                                                                                     1                                                 2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     y                                                                                                                                     the constants c and c to get D .
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          a                                         b                                                                                                                                                              a                                         c                                                                                                                                                                       1                                                 2                                                                         y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     2                                        2                                                                                                                                                               2                                       2
                                                                              M09_BLIT59845_04_SE_C08.QXD  7/9/10  9:50 AM  Page 859
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 Section 8.5 Determinants and Cramer’s Rule                                                                                                                                                                                                                                                                                                                                                                                859
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        EXAMPLE 2 Using Cramer’s Rule to Solve a Linear System
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Use Cramer’s rule to solve the system:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              5x - 4y = 2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 b6x - 5y = 1.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Solution                                                                                                     Because
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           D
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 D                                                                                                                                                        y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       x =                                                      x                            and  y =                                                                                                                ,
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    D                                                                                                                                         D
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        we will set up and evaluate the three determinants D, D , and D .
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           x                                                                       y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          1. D, the determinant in both denominators, consists of the x- and y-coefficients.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   D = `5                                                                                     -4` = 1521-52 - 1621-42 =-25 + 24 =-1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          6                                   -5
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  Because this determinant is not zero,we continue to use Cramer’s rule to solve
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   the system.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           2. D , the determinant in the numerator for x, is obtained by replacing the
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  x
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   x-coefficients in D, 5 and 6,by the constants on the right sides of the equations,
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  2 and 1.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               D = `2                                                                                            -4` = 1221-52 - 1121-42 =-10 + 4 =-6
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              x                                              1                                   -5
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          3. D , the determinant in the numerator for y, is obtained by replacing the
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  y
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   y-coefficients in D, -4 and  -5, by the constants on the right sides of the
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  equations,2 and 1.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           52
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             D = `                                                                                                         `             = 152112 - 162122 = 5 - 12 =-7
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             y                                             61
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          4. Thus,
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                D
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     D                                                        -6                                                                                                                                                                                               y                                         -7
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           x =                                                      x =                                                                        = 6 and  y =                                                                                                                                                                      =                                                        = 7.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        D                                                     -1                                                                                                                                                                                   D                                                     -1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  As always, the solution (6, 7) can be checked by substituting these values into
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  the original equations.The solution set is  516, 726.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         Check Point 2 Use Cramer’srule to solve the system:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            e5x + 4y = 12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         3x - 6y = 24.
                                                                                                                                                                                     3Evaluate a third-order                                                                                                                                                                                                                                                                                                            The Determinant of a 3 : 3 Matrix
                                                                                                                                                                                                                              determinant.                                                                                                                                                                                                                                                                              Associated with every square matrix is a real number called its determinant. The
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        determinant for a 3 * 3 matrix is defined as follows:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      Definition of a Third-Order Determinant
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               a                                        b                                        c
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         1                                        1                                       1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         3     a                                        b                                        c 3 = a b c + b c a + c a b - a b c - b c a - c a b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         2                                        2                                       2                                                        1 2 3                                                                          1 2 3                                                                          1 2 3                                                                             3 2 1                                                                           3 2 1                                                                          3 2 1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               a                                        b                                        c
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         3                                        3                                       3
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