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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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