138x Filetype PDF File size 0.68 MB Source: teachers.dadeschools.net
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
no reviews yet
Please Login to review.