8.1 Matrices and Systems of Equations 8.2 Operations with...

8Matrices andDeterminants

8.1 Matrices and Systems of Equations

8.2 Operations with Matrices

8.3 The Inverse of a Square Matrix

8.4 The Determinant of a Square Matrix

8.5 Applications of Matrices and Determinants

In Mathematics

Matrices are used to model and solve a

variety of problems. For instance, you can

use matrices to solve systems of linear

equations.

In Real Life

Matrices are used to model inventory

levels, electrical networks, investment

portfolios, and other real-life situations.

For instance, you can use a matrix to

model the number of people in the United

States who participate in snowboarding.

(See Exercise 114, page 583.)

IN CAREERS

There are many careers that use matrices. Several are listed below.

Gra

ham

Hey

woo

d/is

tock

phot

o.co

m

• Bank Teller

Exercise 110, page 582

• Political Analyst


• Small Business Owner

Exercises 69–72, pages 606 and 607

• Florist


569

Matrices

In this section, you will study a streamlined technique for solving systems of linear

equations. This technique involves the use of a rectangular array of real numbers called

a matrix. The plural of matrix is matrices.

The entry in the th row and th column is denoted by the double subscript notation

For instance, refers to the entry in the second row, third column. A matrix having

rows and columns is said to be of order If the matrix is square of

order For a square matrix, the entries are the main

diagonal entries.

Order of Matrices

Determine the order of each matrix.

a. b.

c. d.

Solution

a. This matrix has one row and one column. The order of the matrix is

b. This matrix has one row and four columns. The order of the matrix is

c. This matrix has two rows and two columns. The order of the matrix is

d. This matrix has three rows and two columns. The order of the matrix is

Now try Exercise 9.

A matrix that has only one row is called a row matrix, and a matrix that has only

one column is called a column matrix.

3 2.

2 2.

1 4.

1 1.

�5

2

�7

0

�2

4��0

0

0

0�

�1 �3 012��2�

Example 1

a11, a22, a33, . . .m m �or n n�.m � n,m n.n

ma23

a ij.ji

570 Chapter 8 Matrices and Determinants

8.1 MATRICES AND SYSTEMS OF EQUATIONS

What you should learn

• Write matrices and identify theirorders.

• Perform elementary row operationson matrices.

• Use matrices and Gaussian elimination to solve systems of linear equations.

• Use matrices and Gauss-Jordan elimination to solve systems of linear equations.

Why you should learn it

You can use matrices to solve systems of linear equations in two or more variables. For instance, inExercise 113 on page 582, you will use a matrix to find a model for thepath of a ball thrown by a baseballplayer.

Foto

Age

ncy/

Phot

oLib

rary

Definition of Matrix

If and are positive integers, an (read “ by ”) matrix is a rectangular

array

Column 1 Column 2 Column 3 . . . Column n

in which each entry, of the matrix is a number. An matrix has rows

and columns. Matrices are usually denoted by capital letters.n

mm nai j,

�a11

a21

a31...

am1

a12

a22

a32...

am2

a13

a23

a33...

am3

. . .

. . .

. . .

. . .

a1n

a2n

a3n...

amn

�

nmm nnm

Row 1

Row 2

Row 3

Row m

.

.

.

Section 8.1 Matrices and Systems of Equations 571

A matrix derived from a system of linear equations (each written in standard form

with the constant term on the right) is the augmented matrix of the system. Moreover,

the matrix derived from the coefficients of the system (but not including the constant

terms) is the coefficient matrix of the system.

System:

Augmented

Matrix:

Coefficient

Matrix:

Note the use of 0 for the missing coefficient of the -variable in the third equation,

and also note the fourth column of constant terms in the augmented matrix.

When forming either the coefficient matrix or the augmented matrix of a system,

you should begin by vertically aligning the variables in the equations and using zeros

for the coefficients of the missing variables.

Writing an Augmented Matrix

Write the augmented matrix for the system of linear equations.

What is the order of the augmented matrix?

Solution

Begin by rewriting the linear system and aligning the variables.

Next, use the coefficients and constant terms as the matrix entries. Include zeros for the

coefficients of the missing variables.

The augmented matrix has four rows and five columns, so it is a matrix. The

notation is used to designate each row in the matrix. For example, Row 1 is

represented by

Now try Exercise 17.

R1.

Rn

4 5

R1

R2

R3

R4

�1

0

1

2

3

�1

0

4

0

4

�5

�3

�1

2

�6

0

...

...

...

...

9

�2

0

4�

�x � 3y � w � 9

�y � 4z � 2w � �2

x � 5z � 6w � 0

2x � 4y � 3z � 4

�x � 3y � w �

�y � 4z � 2w �

x � 5z � 6w �

2x � 4y � 3z �

9

�2

0

4

Example 2

y

�1

�1

2

�4

3

0

3

�1

�4�

�1

�1

2

�4

3

0

3

�1

�4

......

...

5

�3

6�

�x

�x

2x

� 4y

� 3y

� 3z �

� z �

� 4z �

5

�3

6The vertical dots in an

augmented matrix separate the

coefficients of the linear system

from the constant terms.


Elementary Row Operations

In Section 7.3, you studied three operations that can be used on a system of linear

equations to produce an equivalent system.

1. Interchange two equations.

2. Multiply an equation by a nonzero constant.

3. Add a multiple of an equation to another equation.

In matrix terminology, these three operations correspond to elementary row

operations.An elementary row operation on an augmented matrix of a given system of

linear equations produces a new augmented matrix corresponding to a new (but

equivalent) system of linear equations. Two matrices are row-equivalent if one can be

obtained from the other by a sequence of elementary row operations.

Although elementary row operations are simple to perform, they involve a lot of

arithmetic. Because it is easy to make a mistake, you should get in the habit of noting

the elementary row operations performed in each step so that you can go back and

check your work.


a. Interchange the first and second rows of the original matrix.

Original Matrix New Row-Equivalent Matrix

b. Multiply the first row of the original matrix by


c. Add times the first row of the original matrix to the third row.


Note that the elementary row operation is written beside the row that is changed.


�2R1 � R3→�1

0

0

2

3

�3

�4

�2

13

3

�1

�8��

1

0

2

2

3

1

�4

�2

5

3

�1

�2�

�2

12R1→ �

1

1

5

�2

3

�2

3

�3

1

�1

0

2��

2

1

5

�4

3

�2

6

�3

1

�2

0

2�

12.

R2

R1 ��1

0

2

2

1

�3

0

3

4

3

4

1��

0

�1

2

1

2

�3

3

0

4

4

3

1�

Example 3


1. Interchange two rows.

2. Multiply a row by a nonzero constant.

3. Add a multiple of a row to another row.

TECHNOLOGY

Most graphing utilities can perform elementary row operations on matrices. Consultthe user’s guide for your graphingutility for specific keystrokes.

After performing a row operation, the new row-equivalentmatrix that is displayed on yourgraphing utility is stored in theanswer variable. You should usethe answer variable and not theoriginal matrix for subsequentrow operations.


In Example 3 in Section 7.3, you used Gaussian elimination with back-substitution

to solve a system of linear equations. The next example demonstrates the matrix

version of Gaussian elimination. The two methods are essentially the same. The basic

difference is that with matrices you do not need to keep writing the variables.

Comparing Linear Systems and Matrix Operations

Linear System Associated Augmented Matrix

Add the first equation to the Add the first row to the

second equation. second row

Add times the first equation Add times the first row

to the third equation. to the third row

Add the second equation to the Add the second row to the

third equation. third row

Multiply the third equation by Multiply the third row by

At this point, you can use back-substitution to find x and y.

Substitute 2 for z.

Solve for y.

Substitute for y and 2 for z.

Solve for x.

The solution is and


z � 2.y � �1,x � 1,

x � 1

�1x � 2��1� � 3�2� � 9

y � �1

y � 3�2� � 5

12R3→

�1

0

0

�2

1

0

3

3

1

.

..

.

..

.

..

9

5

2��

x � 2y � 3z � 9

y � 3z � 5

z � 2

�12R3�.1

212.

R2 � R3→�1

0

0

�2

1

0

3

3

2

.

..

.

..

.

..

9

5

4��

x � 2y � 3z �

y � 3z �

2z �

9

5

4

�R2 � R3�.

�2R1 � R3→�1

0

0

�2

1

�1

3

3

�1

.

..

.

..

.

..

9

5

�1��

x � 2y � 3z �

y � 3z �

�y � z �

9

5

�1

��2R1 � R3�.�2�2

R1 � R2→ �1

0

2

�2

1

�5

3

3

5

.

..

.

..

.

..

9

5

17��

x � 2y � 3z �

y � 3z �

2x � 5y � 5z �

9

5

17

�R1 � R2�.

�1

�1

2

�2

3

�5

3

0

5

.

..

.

..

.

..

9

�4

17��

x � 2y

�x � 3y

2x � 5y

� 3z

� 5z

�

�

�

9

�4

17

Example 4

Remember that you should

check a solution by substituting

the values of and into

each equation of the original

system. For example, you can

check the solution to Example 4

as follows.

Equation 1:

�

Equation 2:

�

Equation 3:

�2�1� � 5��1� � 5�2� � 17

�1 � 3��1� � �4

1 � 2��1� � 3�2� � 9

zy,x,

WARNING / CAUTION

Arithmetic errors are often

made when elementary row

operations are performed. Note

the operation you perform in

each step so that you can go

back and check your work.

The last matrix in Example 4 is said to be in row-echelon form. The term echelon

refers to the stair-step pattern formed by the nonzero elements of the matrix. To be in

this form, a matrix must have the following properties.

It is worth noting that the row-echelon form of a matrix is not unique. That is, two

different sequences of elementary row operations may yield different row-echelon

forms. However, the reduced row-echelon form of a given matrix is unique.

Row-Echelon Form

Determine whether each matrix is in row-echelon form. If it is, determine whether the

matrix is in reduced row-echelon form.

a. b.

c. d.

e. f.

Solution

The matrices in (a), (c), (d), and (f) are in row-echelon form. The matrices in

(d) and (f) are in reduced row-echelon form because every column that has a leading 1

has zeros in every position above and below its leading 1. The matrix in (b) is not in

row-echelon form because a row of all zeros does not occur at the bottom of the matrix.

The matrix in (e) is not in row-echelon form because the first nonzero entry in Row 2

is not a leading 1.


Every matrix is row-equivalent to a matrix in row-echelon form. For instance, in

Example 5, you can change the matrix in part (e) to row-echelon form by multiplying its

second row by 12.

�0

0

0

1

0

0

0

1

0

5

3

0��

1

0

0

2

2

0

�3

1

1

4

�1

�3�

�1

0

0

0

0

1

0

0

0

0

1

0

�1

2

3

0��

1

0

0

0

�5

0

0

0

2

1

0

0

�1

3

1

0

3

�2

4

1�

�1

0

0

2

0

1

�1

0

2

2

0

�4��

1

0

0

2

1

0

�1

0

1

4

3

�2�

Example 5


Row-Echelon Form and Reduced Row-Echelon Form

A matrix in row-echelon form has the following properties.

1. Any rows consisting entirely of zeros occur at the bottom of the matrix.

2. For each row that does not consist entirely of zeros, the first nonzero entry

is 1 (called a leading 1).

3. For two successive (nonzero) rows, the leading 1 in the higher row is farther

to the left than the leading 1 in the lower row.

A matrix in row-echelon form is in reduced row-echelon form if every column

that has a leading 1 has zeros in every position above and below its leading 1.


Gaussian Elimination with Back-Substitution

Gaussian elimination with back-substitution works well for solving systems of linear

equations by hand or with a computer. For this algorithm, the order in which the

elementary row operations are performed is important. You should operate from left

to right by columns, using elementary row operations to obtain zeros in all entries

directly below the leading 1’s.


Solve the system

Solution

The matrix is now in row-echelon form, and the corresponding system is

Using back-substitution, you can determine that the solution is

and


w � 3.

z � 1,y � 2,x � �1,

�x � 2y � z

y � z

z

�

� 2w �

� w �

w �

2

�3

�2

3

.

Perform operations on R3

and R4 so third and fourth

columns have leading 1’s.

�113R4→

13R3→ �

1

0

0

0

2

1

0

0

�1

1

1

0

0

�2

�1

1

.

..

.

..

.

..

.

..

2

�3

�2

3�


so second column has

zeros below its leading 1.

6R2 � R4→�1

0

0

0

2

1

0

0

�1

1

3

0

0

�2

�3

�13

.

..

.

..

.

..

.

..

2

�3

�6

�39�


and R4 so first column has

zeros below its leading 1.�2R1 � R3→

�R1 � R4→�1

0

0

0

2

1

0

�6

�1

1

3

�6

0

�2

�3

�1

.

..

.

..

.

..

.

..

2

�3

�6

�21�

Interchange R1 and R2 so

first column has leading

1 in upper left corner.

R2

R1 �1

0

2

1

2

1

4

�4

�1

1

1

�7

0

�2

�3

�1

.

..

.

..

.

..

.

..

2

�3

�2

�19�

Write augmented matrix.�0

1

2

1

1

2

4

�4

1

�1

1

�7

�2

0

�3

�1

.

..

.

..

.

..

.

..

�3

2

�2

�19�

� x

2x

x

�

�

�

y � z

2y � z

4y � z

4y � 7z

� 2w �

�

� 3w �

� w �

�3

2

�2

�19

.

Example 6

The procedure for using Gaussian elimination with back-substitution is

summarized below.

When solving a system of linear equations, remember that it is possible

for the system to have no solution. If, in the elimination process, you obtain a

row of all zeros except for the last entry, it is unnecessary to continue the elimination

process. You can simply conclude that the system has no solution, or is inconsistent.

A System with No Solution

Solve the system

Solution

Write augmented matrix.

Perform row operations.

Perform row operations.

Note that the third row of this matrix consists entirely of zeros except for the last entry.

This means that the original system of linear equations is inconsistent. You can see why

this is true by converting back to a system of linear equations.

Because the third equation is not possible, the system has no solution.


�x � y � 2z �

y � z �

0 �

5y � 7z �

4

2

�2

�11

R2 � R3→ �1

0

0

0

�1

1

0

5

2

�1

0

�7

.

..

.

..

.

..

.

..

4

2

�2

�11�

�R1 � R2→

�2R1 � R3→

�3R1 � R4→�1

0

0

0

�1

1

�1

5

2

�1

1

�7

.

..

.

..

.

..

.

..

4

2

�4

�11�

�1

1

2

3

�1

0

�3

2

2

1

5

�1

.

..

.

..

.

..

.

..

4

6

4

1�

�x

x

2x

3x

�

�

�

y � 2z � 4

� z � 6

3y � 5z � 4

2y � z � 1

.

Example 7



1. Write the augmented matrix of the system of linear equations.

2. Use elementary row operations to rewrite the augmented matrix in

row-echelon form.

3. Write the system of linear equations corresponding to the matrix in

row-echelon form, and use back-substitution to find the solution.


Gauss-Jordan Elimination

With Gaussian elimination, elementary row operations are applied to a matrix to obtain

a (row-equivalent) row-echelon form of the matrix. A second method of elimination,

called Gauss-Jordan elimination, after Carl Friedrich Gauss and Wilhelm Jordan

(1842–1899), continues the reduction process until a reduced row-echelon form is

obtained. This procedure is demonstrated in Example 8.

Gauss-Jordan Elimination

Use Gauss-Jordan elimination to solve the system

Solution

In Example 4, Gaussian elimination was used to obtain the row-echelon form of the

linear system above.

Now, apply elementary row operations until you obtain zeros above each of the

leading 1’s, as follows.

The matrix is now in reduced row-echelon form. Converting back to a system of linear

equations, you have

Now you can simply read the solution, and which can be

written as the ordered triple


The elimination procedures described in this section sometimes result in fractional

coefficients. For instance, in the elimination procedure for the system

you may be inclined to multiply the first row by to produce a leading 1, which will

result in working with fractional coefficients. You can sometimes avoid fractions by

judiciously choosing the order in which you apply elementary row operations.

12

�2x � 5y

3x � 2y

�3x � 3y

� 5z �

� 3z �

�

17

11

�6

�1,�1, 2�.z � 2,y � �1,x � 1,

�xyz

�

�

�

1

�1.

2


and R2 so third column has

zeros above its leading 1.

�9R3 � R1→

�3R3 � R2→ �1

0

0

0

1

0

0

0

1

.

..

.

..

.

..

1

�1

2�


so second column has a

zero above its leading 1.

2R2 � R1→ �1

0

0

0

1

0

9

3

1

.

..

.

..

.

..

19

5

2�

�1

0

0

�2

1

0

3

3

1

.

..

.

..

.

..

9

5

2�

�x � 2y

�x � 3y

2x � 5y

� 3z �

�

� 5z �

9

�4

17

.

Example 8

The advantage of using Gauss-

Jordan elimination to solve a

system of linear equations is

that the solution of the system

is easily found without using

back-substitution, as illustrated

in Example 8.

TECHNOLOGY

For a demonstration of a graphicalapproach to Gauss-Jordan elimination on a matrix,see the Visualizing RowOperations Program available for several models of graphingcalculators at the website for thistext at academic.cengage.com.

2 3


Recall from Chapter 7 that when there are fewer equations than variables in a

system of equations, then the system has either no solution or infinitely many solutions.

A System with an Infinite Number of Solutions

Solve the system.

Solution

The corresponding system of equations is

Solving for and in terms of you have

and

To write a solution of the system that does not use any of the three variables of the

system, let represent any real number and let

Now substitute for in the equations for and

So, the solution set can be written as an ordered triple with the form

where a is any real number. Remember that a solution set of this form represents an

infinite number of solutions. Try substituting values for to obtain a few solutions.

Then check each solution in the original system of equations.


a

��5a � 2, 3a � 1, a�

y � 3z � 1 � 3a � 1

x � �5z � 2 � �5a � 2

y.xza

z � a.

a

y � 3z � 1.x � �5z � 2

z,yx

�x � 5z � 2

y � 3z � �1.

�2R2 � R1→ �1

0

0

1

5

�3

.

..

.

..2

�1��R2→

�1

0

2

1

�1

�3

.

..

.

..0

�1��3R1 � R2→

�1

0

2

�1

�1

3

.

..

.

..0

1�

12R1→ �1

3

2

5

�1

0

.

..

.

..0

1�

�2

3

4

5

�2

0

.

..

.

..0

1�

�2x � 4y

3x � 5y

� 2z � 0

� 1

Example 9

In Example 9, and are

solved for in terms of the third

variable To write a solution

of the system that does not use

any of the three variables of the

system, let represent any real

number and let Then

solve for and The solution

can then be written in terms

of which is not one of the

variables of the system.

a,

y.x

z� a.

a

z.

yx


EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises.8.1

VOCABULARY: Fill in the blanks.

1. A rectangular array of real numbers that can be used to solve a system of linear equations is called a ________.

2. A matrix is ________ if the number of rows equals the number of columns.

3. For a square matrix, the entries are the ________ ________ entries.

4. A matrix with only one row is called a ________ matrix, and a matrix with only one column is called

a ________ matrix.

5. The matrix derived from a system of linear equations is called the ________ matrix of the system.

6. The matrix derived from the coefficients of a system of linear equations is called the ________ matrix of the system.

7. Two matrices are called ________ if one of the matrices can be obtained from the other by a sequence of

elementary row operations.

8. A matrix in row-echelon form is in ________ ________ ________ if every column that has a leading 1 has zeros

in every position above and below its leading 1.

SKILLS AND APPLICATIONS

a33, . . . , anna22,a11,

In Exercises 9–14, determine the order of the matrix.

9. 10.

11. 12.

13. 14.

In Exercises 15–20, write the augmented matrix for thesystem of linear equations.

15. 16.

17. 18.

19. 20.

In Exercises 21–26, write the system of linear equationsrepresented by the augmented matrix. (Use variables and if applicable.)

21. 22.

23.

24.

25.

26.

In Exercises 27–34, fill in the blank(s) using elementary rowoperations to form a row-equivalent matrix.

27. 28.

29. 30.

31. 32.

�1

0

0

0

1

0

6

0

1

1

��

��1

0

0

0

1

0

��2

1

�2

�7�

�1

0

0

0

�1

0

6

0

�1

1

7

3��

1

0

0

5

1

0

4

�2

1

�1

2

�7�

� 1

18

�1

�8�

4��1

0

1

�1

�1��3

18

3

�8

12

4��1

5

1

�2

1

4��14 �

�3

83

6��1

0

4

�

3

�1��3

4

6

�3

8

6��1

2

4

10

3

5�

�6

�1

4

0

2

0

�1

8

�1

7

�10

1

�5

3

6

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33. 34.

In Exercises 35–38, identify the elementary row operation(s)being performed to obtain the new row-equivalent matrix.


35.


36.


37.


38.

39. Perform the sequence of row operations on the matrix.

What did the operations accomplish?

(a) Add times to

(b) Add times to

(c) Add times to

(d) Multiply by

(e) Add times to

40. Perform the sequence of row operations on the matrix.

What did the operations accomplish?

(a) Add to

(b) Interchange and

(c) Add 3 times to

(d) Add times to

(e) Multiply by

(f) Add the appropriate multiples of to and

In Exercises 41–44, determine whether the matrix is in row-echelon form. If it is, determine if it is also in reducedrow-echelon form.

41. 42.

43. 44.

In Exercises 45–48, write the matrix in row-echelon form.(Remember that the row-echelon form of a matrix is notunique.)

45. 46.

47. 48.

In Exercises 49–54, use the matrix capabilities of a graphingutility to write the matrix in reduced row-echelon form.

49. 50.

51.

52.

53. 54.

In Exercises 55–58, write the system of linear equationsrepresented by the augmented matrix. Then use back-substitution to solve. (Use variables and if applicable.)

55. 56. �1

0

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57. 58.

In Exercises 59–62, an augmented matrix that represents a system of linear equations (in variables and if appli-cable) has been reduced using Gauss-Jordan elimination.Write the solution represented by the augmented matrix.

59. 60.

61. 62.

In Exercises 63–84, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

63. 64.

65. 66.

67. 68.

69. 70.

71. 72.

73. 74.

75. 76.

77. 78.

79. 80.

81. 82.

83.

84.

In Exercises 85–90, use the matrix capabilities of a graphingutility to reduce the augmented matrix corresponding to thesystem of equations, and solve the system.

85. 86.

87.

88.

89.

90.

In Exercises 91–94, determine whether the two systems of linear equations yield the same solution. If so, find thesolution using matrices.

91. (a) (b)

92. (a) (b)

93. (a) (b)

94. (a) (b)

In Exercises 95–98, use a system of equations to find thequadratic function that satisfies theequations. Solve the system using matrices.

95.

96. f �1� � 2, f �2� � 9, f �3� � 20

f �1� � 1, f �2� � �1, f �3� � �5

f x! ! ax21 bx 1 c

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y � 2z �

z �

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14

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y � 6z �

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y � 5z � 42

z � 8�x � 4y � 5z �

y � 7z �

z �

27

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8

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y

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x � 3y � 3z � 2w �

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97.

98.

In Exercises 99–102, use a system of equations to find thecubic function that satisfies theequations. Solve the system using matrices.

99. 100.

101. 102.

103. Use the system

to write two different matrices in row-echelon form

that yield the same solution.

104. ELECTRICAL NETWORK The currents in an electrical

network are given by the solution of the system

where and are measured in amperes. Solve the

system of equations using matrices.

105. PARTIAL FRACTIONS Use a system of equations to

write the partial fraction decomposition of the rational

expression. Solve the system using matrices.

106. PARTIAL FRACTIONS Use a system of equations to

write the partial fraction decomposition of the rational

expression. Solve the system using matrices.

107. FINANCE A small shoe corporation borrowed

$1,500,000 to expand its line of shoes. Some of the

money was borrowed at 7%, some at 8%, and some

at 10%. Use a system of equations to determine

how much was borrowed at each rate if the annual

interest was $130,500 and the amount borrowed at

10% was 4 times the amount borrowed at 7%. Solve

the system using matrices.

108. FINANCE A small software corporation borrowed$500,000 to expand its software line. Some of themoney was borrowed at 9%, some at 10%, and someat 12%. Use a system of equations to determine howmuch was borrowed at each rate if the annual interestwas $52,000 and the amount borrowed at 10% was

times the amount borrowed at 9%. Solve the systemusing matrices.

109. TIPS A food server examines the amount of money

earned in tips after working an 8-hour shift. The server

has a total of $95 in denominations of $1, $5, $10, and

$20 bills. The total number of paper bills is 26. The

number of $5 bills is 4 times the number of $10 bills,

and the number of $1 bills is 1 less than twice the

number of $5 bills. Write a system of linear equations

to represent the situation. Then use matrices to find the

number of each denomination.

110. BANKING A bank teller is counting the total amount

of money in a cash drawer at the end of a shift. There

is a total of $2600 in denominations of $1, $5, $10,

and $20 bills The total number of paper bills is 235.

The number of $20 bills is twice the number of $1

bills, and the number of $5 bills is 10 more than the

number of $1 bills. Write a system of linear equations

to represent the situation. Then use matrices to find the

number of each denomination.

In Exercises 111 and 112, use a system of equations to findthe equation of the parabola that passesthrough the points. Solve the system using matrices. Use agraphing utility to verify your results.

111. 112.

113. MATHEMATICAL MODELING A video of the path

of a ball thrown by a baseball player was analyzed

with a grid covering the TV screen. The tape was

paused three times, and the position of the ball was

measured each time. The coordinates obtained are

shown in the table. ( and are measured in feet.)yx

−8

(2, 8)(3, 5)

(1, 9)

x

8

y

−4 8 12

12

−8

24

(1, 8)

(2, 13)

(3, 20)

x

y

−4 4 8 12

y ! ax 2" bx " c

212

8x2

�x � 1�2�x � 1� �A

x � 1�

B

x � 1�

C

�x � 1�2

4x 2

�x � 1�2�x � 1��

A

x � 1�

B

x � 1�

C

�x � 1�2

I3I1, I2,

�I1 � I2

3I1 � 4I2

I2

� I3 � 0

� 18

� 3I3 � 6

� x � 3y � z � 3

x � 5y � 5z � 1

2x � 6y � 3z � 8

f �2� � 7f �2� � �7

f �1� � 1f �1� � �4

f ��1� � �5f ��1� � 2

f ��2� � �17f ��2� � �7

f �3� � 44f �3� � 11

f �2� � 16f �2� � 1

f �1� � 4f �1� � �1

f ��1� � 4f ��1� � �5

f x! ! ax31 bx2

1 cx 1 d

f ��2� � �3, f �1� � �3, f �2� � �11

f ��2� � �15, f ��1� � 7, f �1� � �3

Horizontal distance, x Height, y

0

15

30

5.0

9.6

12.4


(a) Use a system of equations to find the equation of

the parabola that passes through

the three points. Solve the system using matrices.

(b) Use a graphing utility to graph the parabola.

(c) Graphically approximate the maximum height of the

ball and the point at which the ball struck the ground.

(d) Analytically find the maximum height of the ball

and the point at which the ball struck the ground.

(e) Compare your results from parts (c) and (d).

114. DATA ANALYSIS: SNOWBOARDERS The table

shows the numbers of people (in millions) in the

United States who participated in snowboarding in

selected years from 2003 to 2007. (Source: National

Sporting Goods Association)

(a) Use a system of equations to find the equation of

the parabola that passes through

the points. Let represent the year, with

corresponding to 2003. Solve the system using

matrices.

(b) Use a graphing utility to graph the parabola.

(c) Use the equation in part (a) to estimate the number

of people who participated in snowboarding in

2009. Does your answer seem reasonable? Explain.

(d) Do you believe that the equation can be used for

years far beyond 2007? Explain.

NETWORK ANALYSIS In Exercises 115 and 116, answerthe questions about the specified network. (In a network it isassumed that the total flow into each junction is equal to thetotal flow out of each junction.)

115. Water flowing through a network of pipes (in thousands

of cubic meters per hour) is shown in the figure.

(a) Solve this system using matrices for the water flow

represented by

(b) Find the network flow pattern when and

(c) Find the network flow pattern when and

116. The flow of traffic (in vehicles per hour) through a

network of streets is shown in the figure.

(a) Solve this system using matrices for the traffic

flow represented by

(b) Find the traffic flow when and

(c) Find the traffic flow when and

EXPLORATION

TRUE OR FALSE? In Exercises 117 and 118, determinewhether the statement is true or false. Justify your answer.

117. is a matrix.

118. The method of Gaussian elimination reduces a matrix

until a reduced row-echelon form is obtained.

119. THINK ABOUT IT The augmented matrix below

represents system of linear equations (in variables

and ) that has been reduced using Gauss-Jordan elim-

ination. Write a system of equations with nonzero

coefficients that is represented by the reduced matrix.

(There are many correct answers.)

120. THINK ABOUT IT

(a) Describe the row-echelon form of an augmented

matrix that corresponds to a system of linear

equations that is inconsistent.

(b) Describe the row-echelon form of an augmented

matrix that corresponds to a system of linear

equations that has an infinite number of solutions.

121. Describe the three elementary row operations that can

be performed on an augmented matrix.

123. What is the relationship between the three elementary

row operations performed on an augmented matrix and

the operations that lead to equivalent systems of

equations?

�100

0

1

0

3

4

0

�2

1

0�

z

y,x,

4 2� 5

�1

0

3

�2

�6

7

0�

x3 � 0.x2 � 150

x3 � 50.x2 � 200

xi, i � 1, 2, . . . , 5.

350200

x1

x2 x4x3

x5

300 150

x6 � 500.

x5 � 400

x 7� 0.

x6 � 0

xi, i � 1, 2, . . . , 7.

600 500

x1

x6 x7

x2

x3 x4 x5

600 500

t � 3t

y � at2 � bt � c

y

y � ax2 � bx � c

Year Number, y

2003

2005

2007

6.3

6.0

5.1

122. CAPSTONE In your own words, describe the

difference between a matrix in row-echelon form and

a matrix in reduced row-echelon form. Include an

example of each to support your explanation.


8.2 OPERATIONS WITH MATRICES


• Decide whether two matrices areequal.

• Add and subtract matrices and multiply matrices by scalars.

• Multiply two matrices.

• Use matrix operations to model and solve real-life problems.


Matrix operations can be usedto model and solve real-life problems.For instance, in Exercise 76 on page598, matrix operations are used toanalyze annual health care costs.

© R

oyal

ty-F

ree/

Cor

bis

Equality of Matrices

In Section 8.1, you used matrices to solve systems of linear equations. There is a rich

mathematical theory of matrices, and its applications are numerous. This section and

the next two introduce some fundamentals of matrix theory. It is standard mathematical

convention to represent matrices in any of the following three ways.

Two matrices and are equal if they have the same order

and for and In other words, two matrices are equal if

their corresponding entries are equal.

Equality of Matrices

Solve for and in the following matrix equation.

Solution

Because two matrices are equal only if their corresponding entries are equal, you can

conclude that

and

Now try Exercise 7.

Be sure you see that for two matrices to be equal, they must have the same order

and their corresponding entries must be equal. For instance,

but �2

3

0

�1

4

0� � �2

3

�1

4�.� 2

4

�112� � �2

2

�1

0.5�

a22 � 0.a21 � �3,a12 � �1,a11 � 2,

�a11

a21

a12

a22� � � 2

�3

�1

0�a22a11, a12, a21,

Example 1

1 ! j ! n.1 ! i ! maij � bij

�m n�B � �bij �A � �aij�

Representation of Matrices

1. A matrix can be denoted by an uppercase letter such as A, B, or C.

2. A matrix can be denoted by a representative element enclosed in brackets,

such as or

3. A matrix can be denoted by a rectangular array of numbers such as

A � �aij � � �a11

a21

a31

...

am1

a12

a22

a32

...

am2

a13

a23

a33

...

am3

. . .

. . .

. . .

. . .

a1n

a2n

a3n

...

amn

�.

�cij�.�bij�,�aij �,

Section 8.2 Operations with Matrices 585

Definition of Matrix Addition

If and are matrices of order their sum is the

matrix given by

The sum of two matrices of different orders is undefined.

A � B � �aij � bij � .

m nm n,B � �bij�A � �aij�

Matrix Addition and Scalar Multiplication

In this section, three basic matrix operations will be covered. The first two are matrix

addition and scalar multiplication. With matrix addition, you can add two matrices (of

the same order) by adding their corresponding entries.

Addition of Matrices

a.

b.

c.

d. The sum of

and

is undefined because is of order and B is of order

Now try Exercise 13(a).

In operations with matrices, numbers are usually referred to as scalars. In this text,

scalars will always be real numbers. You can multiply a matrix by a scalar by

multiplying each entry in by c.A

cA

3 2.3 3A

B � �0

�1

2

1

3

4�

A � �2

4

3

1

0

�2

0

�1

2�

�1

�3

�2� � �

�1

3

2� � �

0

0

0�

�0

1

1

2

�2

3� � �0

0

0

0

0

0� � �0

1

1

2

�2

3�

� � 0

�1

5

3�

��1

0

2

1� � � 1

�1

3

2� � ��1 � 1

0 � ��1�2 � 3

1 � 2�

Example 2

HISTORICAL NOTE

Arthur Cayley (1821–1895), aBritish mathematician, invented

matrices around 1858. Cayleywas a Cambridge University

graduate and a lawyer by profession. His groundbreaking

work on matrices was begun as he studied the theory of

transformations. Cayley also wasinstrumental in the development

of determinants. Cayley and two American mathematicians,Benjamin Peirce (1809–1880)and his son Charles S. Peirce

(1839–1914), are credited withdeveloping “matrix algebra.”

The

Gra

nger

Col

lect

ion

Definition of Scalar Multiplication

If is an matrix and is a scalar, the scalar multiple of by is

the matrix given by

cA � �caij� .

m n

cAcm nA � �aij�

The symbol represents the negation of which is the scalar product

Moreover, if and are of the same order, then represents the sum of and

That is,

Subtraction of matrices

The order of operations for matrix expressions is similar to that for real

numbers. In particular, you perform scalar multiplication before matrix addition and

subtraction, as shown in Example 3(c).

Scalar Multiplication and Matrix Subtraction

For the following matrices, find (a) , (b) and (c)

and

Solution

a. Scalar multiplication

Multiply each entry by 3.

Simplify.

b. Definition of negation

Multiply each entry by

c. Matrix subtraction

Subtract corresponding entries.

Now try Exercise 13(b), (c), and (d).

It is often convenient to rewrite the scalar multiple by factoring out of every

entry in the matrix. For instance, in the following example, the scalar has been

factored out of the matrix.

�12

52

�32

12� � �

12�1�12�5�

12��3�

12�1�� 1

2�1

5

�3

1�

12

ccA

� �4

�10

7

6

4

0

12

�6

4�

3A � B � �6

�9

6

6

0

3

12

�3

6� � �

2

1

�1

0

�4

3

0

3

2�

�1.� ��2

�1

1

0

4

�3

0

�3

�2�

�B � ��1��2

1

�1

0

�4

3

0

3

2�

� �6

�9

6

6

0

3

12

�3

6�

� �3�2�

3��3�3�2�

3�2�3�0�3�1�

3�4�3��1�

3�2��

3A � 3�2

�3

2

2

0

1

4

�1

2�

B � �2

1

�1

0

�4

3

0

3

2�A � �

2

�3

2

2

0

1

4

�1

2�

3A � B.�B,3A

Example 3

A � B � A � ��1�B.

��1�B.

AA � BBA

��1�A.A,�A



The properties of matrix addition and scalar multiplication are similar to those of

addition and multiplication of real numbers.

Note that the Associative Property of Matrix Addition allows you to write

expressions such as without ambiguity because the same sum occurs no

matter how the matrices are grouped. This same reasoning applies to sums of four or

more matrices.

Addition of More than Two Matrices

By adding corresponding entries, you obtain the following sum of four matrices.


Using the Distributive Property

Perform the indicated matrix operations.

Solution


In Example 5, you could add the two matrices first and then multiply the matrix

by 3, as follows. Notice that you obtain the same result.

3��2

4

0

1� � �4

3

�2

7� � 3�2

7

�2

8� � � 6

21

�6

24�

� � 6

21

�6

24�

� ��6

12

0

3� � �12

9

�6

21�

3��2

4

0

1� � �4

3

�2

7� � 3��2

4

0

1� � 3�4

3

�2

7�

3��2

4

0

1� � �4

3

�2

7�

Example 5

�1

2

�3� � �

�1

�1

2� � �

0

1

4� � �

2

�3

�2� � �

2

�1

1�

Example 4

A � B � C

Properties of Matrix Addition and Scalar Multiplication

Let and be matrices and let and be scalars.

1. Commutative Property of Matrix Addition

2. Associative Property of Matrix Addition

3. Associative Property of Scalar Multiplication

4. Scalar Identity Property

5. Distributive Property

6. Distributive Property�c � d�A � cA � dA

c�A � B� � cA � cB

1A � A

�cd �A � c�dA)

A � �B � C� � �A � B� � C

A � B � B � A

dcm nCB,A,

You can review the properties

of addition and multiplication

of real numbers (and other

properties of real numbers) in

Appendix A.1.

TECHNOLOGY

Most graphing utilities have thecapability of performing matrixoperations. Consult the user’sguide for your graphing utility for specific keystrokes. Try using a graphing utility to find thesum of the matrices

and

B ! �#1

2

4

#5�.

A ! � 2

#1

#3

0�


One important property of addition of real numbers is that the number 0 is the

additive identity. That is, for any real number For matrices, a similar

property holds. That is, if is an matrix and is the zero matrix

consisting entirely of zeros, then

In other words, is the additive identity for the set of all matrices. For

example, the following matrices are the additive identities for the sets of all and

matrices.

and

zero matrix zero matrix

The algebra of real numbers and the algebra of matrices have many similarities. For

example, compare the following solutions.

Real Numbers Matrices

(Solve for x.) (Solve for X.)

The algebra of real numbers and the algebra of matrices also have important

differences, which will be discussed later.

Solving a Matrix Equation

Solve for in the equation where

and

Solution

Begin by solving the matrix equation for to obtain

Now, using the matrices and you have

Substitute the matrices.

Subtract matrix from matrix

Multiply the matrix by


13.� ��

43

23

2

�23�.

B.A�1

3��4

2

6

�2�

X �1

3��3

2

4

1� � �1

0

�2

3� B,A

X �1

3�B � A�.

3X � B � A

X

B � ��3

2

4

1�.A � �1

0

�2

3�3X � A � B,X

Example 6

X � B � Ax � b � a

X � O � B � Ax � 0 � b � a

X � A � ��A� � B � ��A�x � a � ��a� � b � ��a�

X � A � Bx � a � b

m n

2 22 3

O � �0

0

0

0�O � �0

0

0

0

0

0�2 2

2 3

m nO

A � O � A.

m nOm nA

c.c � 0 � c

WARNING / CAUTION

Remember that matrices are

denoted by capital letters. So,

when you solve for you are

solving for a matrix that makes

the matrix equation true.

X,


Matrix Multiplication

Another basic matrix operation is matrix multiplication.At first glance, the definition

may seem unusual. You will see later, however, that this definition of the product of two

matrices has many practical applications.

The definition of matrix multiplication indicates a row-by-column multiplication,

where the entry in the th row and th column of the product is obtained by

multiplying the entries in the th row of by the corresponding entries in the th

column of and then adding the results. So for the product of two matrices to be

defined, the number of columns of the first matrix must equal the number of rows of

the second matrix. The general pattern for matrix multiplication is as follows.

Finding the Product of Two Matrices

Find the product using and

Solution

To find the entries of the product, multiply each row of by each column of


� ��9

�4

�15

1

6

10�

� ��1��3� � ��3��4�

�4��3� � ��2��4��5��3� � ��0��4�

��1��2� � ��3��1��4��2� � ��2��1��5��2� � ��0��1��

AB � ��1

4

5

3

�2

0��3

�4

2

1�

B.A

B � ��3

�4

2

1�.A � ��1

4

5

3

�2

0�AB

Example 7

B

jAi

ABji

Definition of Matrix Multiplication

If is an matrix and is an matrix, the product

is an matrix

where ci j � ai1b1j � ai2b2 j � ai3b3j �. . . � ainbnj .

AB � �cij �

m p

ABn pB � �bij�m nA � �aij�

ai1b1j � ai2b2j � ai3b3j �. . . � ainbnj � cij

�c11

c21

.

..

ci1

.

..

cm1

c12

c22

.

..

ci2

.

..

cm2

. . .

. . .

. . .

. . .

c1j

c2j

.

..

cij

.

..

cmj

. . .

. . .

. . .

. . .

c1p

c2p

.

..

cip

.

..

cmp

��b11

b21

b31

.

..

bn1

b12

b22

b32

.

..

bn2

. . .

. . .

. . .

. . .

b1j

b2j

b3j

.

..

bnj

. . .

. . .

. . .

. . .

b1p

b2p

b3p

.

..

bnp

� ��a11

a21

a31...

ai1...

am1

a12

a22

a32...

ai2...

am2

a13

a23

a33...

ai3...

am3

. . .

. . .

. . .

. . .

. . .

a1n

a2n

a3n...

ain...

amn

�

In Example 7, the product is

defined because the number of

columns of is equal to the

number of rows of Also, note

that the product has order

3 2.

AB

B.

A

AB

Be sure you understand that for the product of two matrices to be defined, the number

of columns of the first matrix must equal the number of rows of the second matrix. That

is, the middle two indices must be the same. The outside two indices give the order of

the product, as shown below.

Finding the Product of Two Matrices

Find the product where

and

Solution

Note that the order of is and the order of is So, the product has

order


Patterns in Matrix Multiplication

a.

b.

c. The product for the following matrices is not defined.

and


3 43 2

B � ��2

0

2

3

1

�1

1

�1

0

4

2

1�A � �

�2

1

1

1

�3

4�

AB

3 13 13 3

�10

�5

�9��

1

2

�3��

6

3

1

2

�1

4

0

2

6�

2 22 22 2

� 3

�2

4

5��1

0

0

1�� 3

�2

4

5�

Example 9

� ��5

�3

7

6�

1�4� � 0�0� � 3�1�2�4� � ��1��0� � ��2��1�� 1��2� � 0�1� � 3��1�

2��2� � ��1��1� � ��2��1�

AB � �1

2

0

�1

3

�2��2

1

�1

4

0

1�

2 2.

AB3 2.B2 3A

B � ��2

1

�1

4

0

1�.A � �1

2

0

�1

3

�2�

AB

Example 8

Equal

Order of AB

m pn pm n

AB�B A



Patterns in Matrix Multiplication

a.


In Example 10, note that the two products are different. Even if both and

are defined, matrix multiplication is not, in general, commutative. That is, for most

matrices, This is one way in which the algebra of real numbers and the

algebra of matrices differ.

AB � BA.

BAAB

1 13 11 3

� �1��2

�1

1��1 �2 �3�

Example 10

If A is an matrix, the identity matrix has the property that and

For example,

and

IA � A�1

0

0

0

1

0

0

0

1��

3

1

�1

�2

0

2

5

4

�3� � �

3

1

�1

�2

0

2

5

4

�3� .

AI � A�3

1

�1

�2

0

2

5

4

�3��

1

0

0

0

1

0

0

0

1� � �

3

1

�1

�2

0

2

5

4

�3�

InA � A.

AIn � An n

Properties of Matrix Multiplication

Let and be matrices and let be a scalar.

1. Associative Property of Matrix Multiplication



4. Associative Property of Scalar Multiplicationc�AB� � �cA�B � A�cB�

�A � B)C � AC � BC

A�B � C� � AB � AC

A�BC � � �AB�C

cCB,A,

Definition of Identity Matrix

The matrix that consists of 1’s on its main diagonal and 0’s elsewhere is

called the identity matrix of order and is denoted by

Identity matrix

Note that an identity matrix must be square. When the order is understood to be

you can denote simply by I.Inn n,

In � �1

0

0...

0

0

1

0...

0

0

0

1...

0

. . .

. . .

. . .

. . .

0

0

0...

1

�.

n n

n n

b.

3 31 33 1

� �2

�1

1

�4

2

�2

�6

3

�3��1 �2 �3��

2

�1

1�

Applications

Matrix multiplication can be used to represent a system of linear equations. Note how

the system

can be written as the matrix equation where is the coefficient matrix of the

system, and and are column matrices.

A X B

Solving a System of Linear Equations

Consider the following system of linear equations.

a. Write this system as a matrix equation,

b. Use Gauss-Jordan elimination on the augmented matrix to solve for the

matrix

Solution

a. In matrix form, the system can be written as follows.

b. The augmented matrix is formed by adjoining matrix to matrix

Using Gauss-Jordan elimination, you can rewrite this equation as

So, the solution of the system of linear equations is and

and the solution of the matrix equation is


X � �x1x2x3� � �

�1

2

1�.

x3 � 1,x2 � 2,x1 � �1,

�I X� � �1

0

0

0

1

0

0

0

1

...

...

...

�1

2

1�.

�A B� � �1

0

2

�2

1

3

1

2

�2

..

.

..

.

..

.

�4

4

2�

A.B

�1

0

2

�2

1

3

1

2

�2� �

x1x2x3� � �

�4

4

2�

AX � B,

X.

�A B�AX � B.

�x1

2x1

� 2x2x2

� 3x2

� x3 �

� 2x3 �

� 2x3 �

�4

4

2

Example 11

�

�b1

b2

b3��

x1

x2

x3��

a11

a21

a31

a12

a22

a32

a13

a23

a33�

BX

AAX � B,

�a11x1 � a12x2 � a13x3 � b1

a21x1 � a22x2 � a23x3 � b2

a31x1 � a32x2 � a33x3 � b3


The notation represents

the augmented matrix formed

when matrix is adjoined

to matrix The notation

represents the reduced

row-echelon form of the

augmented matrix that yields

the solution of the system.

�I ... X�A.

B

�A ... B�

The column matrix is also

called a constant matrix. Its

entries are the constant terms

in the system of equations.

B


Softball Team Expenses

Two softball teams submit equipment lists to their sponsors.

Women’s Team Men’s Team

Bats 12 15

Balls 45 38

Gloves 15 17

Each bat costs $80, each ball costs $6, and each glove costs $60. Use matrices to find

the total cost of equipment for each team.

Solution

The equipment lists and the costs per item can be written in matrix form as

and

The total cost of equipment for each team is given by the product

So, the total cost of equipment for the women’s team is $2130 and the total cost of

equipment for the men’s team is $2448.


� �2130 2448�.

� �80�12� � 6�45� � 60�15� 80�15� � 6�38� � 60�17��

CE � �80 6 60��12

45

15

15

38

17�

C � �80 6 60� .

E � �12

45

15

15

38

17�

CE

Example 12

Problem Posing Write a matrix multiplication application problem that uses the matrix

Exchange problems with another student in your class. Form the matrices that represent the problem, and solve the problem. Interpret your solution in the contextof the problem. Check with the creator of the problem to see if you are correct.Discuss other ways to represent and/or approach the problem.

A ! [20

17

42

30

33

50] .

CLASSROOM DISCUSSION

Notice in Example 12 that you

cannot find the total cost using

the product because is

not defined. That is, the number

of columns of (2 columns)

does not equal the number of

rows of (1 row).C

E

ECEC



VOCABULARY

In Exercises 1–4, fill in the blanks.

1. Two matrices are ________ if all of their corresponding entries are equal.

2. When performing matrix operations, real numbers are often referred to as ________.

3. A matrix consisting entirely of zeros is called a ________ matrix and is denoted by ________.

4. The matrix consisting of 1’s on its main diagonal and 0’s elsewhere is called the ________

matrix of order

In Exercises 5 and 6, match the matrix property with the correct form. and are matrices of order and and are scalars.

5. (a) (i) Distributive Property

(b) (ii) Commutative Property of Matrix Addition

(c) (iii) Scalar Identity Property

(d) (iv) Associative Property of Matrix Addition

(e) (v) Associative Property of Scalar Multiplication

6. (a) (i) Distributive Property

(b) (ii) Additive Identity of Matrix Addition

(c) (iii) Associative Property of Matrix Multiplication

(d) (iv) Associative Property of Scalar Multiplication


A�BC� � �AB�CA�B � C� � AB � AC

c�AB� � A�cB�A � O � A

A � B � B � A

�cd�A � c�dA��c � d�A � cA � dA

A � �B � C� � �A � B� � C

1A � A

dcm n,CB,A,

n n.

n n

In Exercises 7–10, find and

7. 8.

9.

10.

In Exercises 11–18, if possible, find (a) (b) (c ) and (d)

11.

12.

13.

14.

15.

16.

17.

18.

In Exercises 19–24, evaluate the expression.

19.

20.

21. 4��400

2

1

3� � �231

�6

�2

0� � 6

�1

8

0� � � 0

�3

5

�1� � ��112�7

�1��53

0

�6� � � 7

�2

1

�1� � ��1014�8

6�

B � ��4 6 2�A � �3

2

�1�,

B � �84�1

�3�A � � 6

�1

0

�4

3

0�,

B � ��3

2

10

3

0

5

�4

�9

2

1

1

�7

�1

�4

�2�A � �

�1

3

5

0

�4

4

�2

4

8

�1

0

2

�1

�6

0�,

B � � 1

�6

0

8

�1

2

1

�3

0

�7�

A � �415

2

�1

�2

3

�1

4

0�,

B � ��2�3

0

4

�5

�7�A � �10�1

6

3

9�,

B � �1

�1

1

6

�5

10�A � �

8

2

�4

�1

3

5�,B � ��34

�2

2�A � �122

1�,B � � 2

�1

�1

8�A � �12�1

�1�,3A ! 2B.3A,

A ! B,A 1 B,

�x � 2

1

7

8

2y

�2

�3

2x

y � 2� � �

2x � 6

1

7

8

18

�2

�3

�8

11�

�16

�3

0

4

13

2

5

15

4

4

6

0� � �

16

�3

0

4

13

2

2x � 1

15

3y � 5

4

3x

0�

��5yx

8� � ��51213

8�� x7�2

y � � ��47�2

22�y.x


22.

23.

24.

In Exercises 25–28, use the matrix capabilities of a graphingutility to evaluate the expression. Round your results to threedecimal places, if necessary.

25.

26.

27.

28.

In Exercises 29–32, solve for in the equation, given

and

29. 30.

31. 32.

In Exercises 33–40, if possible, find and state the order ofthe result.

33.

34.

35.

36.

37.

38.

39.

40.

In Exercises 41–46, use the matrix capabilities of a graphingutility to find if possible.

41.

42.

43.

44.

45.

46.

In Exercises 47–52, if possible, find (a) (b) and (c) (Note: )

47.

48.

49.

50.

51.

52. A � �3 2 1� , B � �2

3

0�

B � �1 1 2�A � �7

8

�1�,

B � � 1

�3

3

1�A � �11�1

1�,B � �13

�3

1�A � �31�1

3�,A � � 6

�2

3

�4�, B � ��220

4�

B � � 2

�1

�1

8�A � �142

2�,A2" AA.

A2.BA,AB,

A � �16

�4

�9

�18

13

21�, B � ��77

20

15

�1

26�

B � �5240�85

�35

27

60

45

82�

A � � 9

100

10

�50

�38

250

18

75�,

A � ��2

21

13

4

5

2

8

6

6�, B � �

2

�7

32

0.5

0

15

14

1.6�

B � �3

24

16

8

1

15

10

�4

6

14

21

10�A � �

�3

�12

5

8

15

�1

�6

9

1

8

6

5�,

A � �11

14

6

�12

10

�2

4

12

9�, B � �

12

�5

15

10

12

16�

A � �7

�2

10

5

5

�4

�4

1

�7�, B � �

2

8

�4

�2

1

2

3

4

�8�

AB,

B � �146

2�A � �160

13

3

8

�2

�17�,B � �6 �2 1 6�A � �1012�,

A � �0

0

0

0

0

0

5

�3

4�, B � �

6

8

0

�11

16

0

4

4

0�

A � �5

0

0

0

�8

0

0

0

7�, B � �

15

0

0

0

�18

0

0

012

�A � �

1

0

0

0

4

0

0

0

�2�, B � �

3

0

0

0

�1

0

0

0

5�

A � ��1

�4

0

6

5

3�, B � �20

3

9�

A � �0

6

7

�1

0

�1

2

3

8�, B � �

2

4

1

�1

�5

6�

A � �2

�3

1

1

4

6�, B � �

0

4

8

�1

0

�1

0

2

7�

AB

2A � 4B � �2X2X � 3A � B

2X � 2A � BX � 3A � 2B

B " [0

2

!4

3

0

!1].A " [

!2

1

3

!1

0

!4]

X

��10

�20

12

15

10

4� � 18��

�13

7

6

11

0

9� � �

�3

�3

�14

13

8

15�

��3.211

�1.004

0.055

6.829

4.914

�3.889� � �

�1.630

5.256

�9.768

�3.090

8.335

4.251�

55�� 14

�22

�11

19� � ��221320

6�

3

7�2

�1

5

�4� � 6��3

2

0

2�

��4

�2

9

11

�1

3� � 16��

�5

3

0

�1

4

13� � �

7

�9

6

5

�1

�1�

�3��07�3

2� � ��683

1� � 2�4

7

�4

�9�

12��5 �2 4 0� � �14 6 �18 9��


In Exercises 53–56, evaluate the expression. Use the matrixcapabilities of a graphing utility to verify your answer.

53.

54.

55.

56.

In Exercises 57–64, (a) write the system of linear equations as a matrix equation, and (b) use Gauss-Jordanelimination on the augmented matrix to solve for thematrix

57. 58.

59. 60.

61.

62.

63.

64.

65. MANUFACTURING A corporation has three factories,

each of which manufactures acoustic guitars and

electric guitars. The number of units of guitars produced

at factory in one day is represented by in the matrix

Find the production levels if production is increased by

20%.

66. MANUFACTURING A corporation has four factories,

each of which manufactures sport utility vehicles and

pickup trucks. The number of units of vehicle produced

at factory in one day is represented by in the matrix

Find the production levels if production is increased by

10%.

67. AGRICULTURE A fruit grower raises two crops,

apples and peaches. Each of these crops is sent to three

different outlets for sale. These outlets are The Farmer’s

Market, The Fruit Stand, and The Fruit Farm. The

numbers of bushels of apples sent to the three outlets are

125, 100, and 75, respectively. The numbers of bushels

of peaches sent to the three outlets are 100, 175, and

125, respectively. The profit per bushel for apples is

$3.50 and the profit per bushel for peaches is $6.00.

(a) Write a matrix that represents the number of

bushels of each crop that are shipped to each outlet

State what each entry of the matrix represents.

(b) Write a matrix that represents the profit per

bushel of each fruit. State what each entry of the

matrix represents.

(c) Find the product and state what each entry of the

matrix represents.

68. REVENUE An electronics manufacturer produces three

models of LCD televisions, which are shipped to two

warehouses. The numbers of units of model that are

shipped to warehouse are represented by in the matrix

The prices per unit are represented by the matrix

Compute and interpret the result.

69. INVENTORY A company sells five models of computers

through three retail outlets. The inventories are repre-

sented by

Model

A B C D E

The wholesale and retail prices are represented by

Price

Wholesale Retail

Compute and interpret the result.ST

A

B

C

D

E

ModelT � �$840

$1200

$1450

$2650

$3050

$1100

$1350

$1650

$3000

$3200�

T.

1

2

3 OutletS � �3

0

4

2

2

2

2

3

1

3

4

3

0

3

2�

S.

BA

B � �$699.95 $899.95 $1099.95�.

A � �5,000

6,000

8,000

4,000

10,000

5,000�.

aijj

i

BA

bij

B

aijj.

i

A

A � �100

40

90

20

70

60

30

60�.

aijj

i

A � �70

35

50

100

25

70� .

aijj

�x1 �

x1 �

x2

3x2

�6x2

�

�

4x3 �

�

5x3 �

17

�11

40

�x1 �

�3x1 �

5x2

x2

�2x2

� 2x3 �

� x3 �

� 5x3 �

�20

8

�16

� x1

�x1

x1

�

�

�

x2

2x2

x2

�

�

3x3

x3

�

�

�

�1

1

2

�x1 � 2x2 � 3x3 �

�x1 � 3x2 � x3 �

2x1 � 5x2 � 5x3 �

9

�6

17

��4x1

x1

�

�

9x2

3x2

�

�

�13

12��2x1

6x1

�

�

3x2

x2

�

�

�4

�36

�2x1

x1

�

�

3x2

4x2

�

�

5

10� �x1

�2x1

�

�

x2

x2

�

�

4

0

X.[A B]

AX " B,

�3

�1

5

7��5 �6� � �7 �1� � ��8 9��

�0

4

2

1

�2

2��4

0

�1

0

�1

2� � �

�2

�3

0

3

5

�3�

�3��6

1

5

�2

�1

0��0

�1

4

3

�3

1�

�3

0

1

�2��1

�2

0

2��1

2

0

4�


70. VOTING PREFERENCES The matrix

From

R D I

is called a stochastic matrix. Each entry

represents the proportion of the voting population that

changes from party to party and represents the

proportion that remains loyal to the party from one

election to the next. Compute and interpret

71. VOTING PREFERENCES Use a graphing utility to

find and for the matrix given in

Exercise 70. Can you detect a pattern as is raised to

higher powers?

72. LABOR/WAGE REQUIREMENTS A company that

manufactures boats has the following labor-hour and

wage requirements.

Labor per boat

Department

Cutting AssemblyPackaging

Wages per hour

Plant

A B


73. PROFIT At a local dairy mart, the numbers of gallons

of skim milk, 2% milk, and whole milk sold over the

weekend are represented by

Skim 2% Wholemilk milk milk

The selling prices (in dollars per gallon) and the profits

(in dollars per gallon) for the three types of milk sold by

the dairy mart are represented by

Selling Profitprice

(a) Compute and interpret the result.

(b) Find the dairy mart’s total profit from milk sales for

the weekend.

74. PROFIT At a convenience store, the numbers of

gallons of 87-octane, 89-octane, and 93-octane gasoline

sold over the weekend are represented by

Octane

87 89 93

The selling prices (in dollars per gallon) and the profits

(in dollars per gallon) for the three grades of gasoline

sold by the convenience store are represented by

Selling Profitprice

(a) Compute and interpret the result.

(b) Find the convenience store’s profit from gasoline

sales for the weekend.

75. EXERCISE The numbers of calories burned by indi-

viduals of different body weights performing different

types of aerobic exercises for a 20-minute time period

are shown in matrix

Calories burned

120-lb 150-lbperson person

(a) A 120-pound person and a 150-pound person bicycled

for 40 minutes, jogged for 10 minutes, and walked for

60 minutes. Organize the time they spent exercising in

a matrix

(b) Compute and interpret the result.BA

B.

Bicycling

Jogging

Walking

A � �109

127

64

136

159

79�

A.

AB

87

89

93 OctaneB � �

$2.00

$2.10

$2.20

$0.08

$0.09

$0.10�

B.

Friday

Saturday

Sunday

A � �580

560

860

840

420

1020

320

160

540�

A.

AB

Skim milk

2% milk

Whole milk

B � �$3.45

$3.65

$3.85

$1.20

$1.30

$1.45�

B.

Friday

Saturday

Sunday

A � �40

60

76

64

82

96

52

76

84�

A.

ST

Cutting

Assembly

Packaging

DepartmentT � �$15

$12

$11

$13

$11

$10�

Small

Medium

Large

Boat sizeS � �1.0 h

1.6 h

2.5 h

0.5 h

1.0 h

2.0 h

0.2 h

0.2 h

1.4 h�

P

P8P7,P6,P5,P4,P3,

P2.

piij,i

pij �i � j�

R

D

I ToP � �

0.6

0.2

0.2

0.1

0.7

0.2

0.1

0.1

0.8�


76. HEALTH CARE The health care plans offered this

year by a local manufacturing plant are as follows. For

individuals, the comprehensive plan costs $694.32, the

HMO standard plan costs $451.80, and the HMO Plus

plan costs $489.48. For families, the comprehensive

plan costs $1725.36, the HMO standard plan costs

$1187.76, and the HMO Plus plan costs $1248.12. The

plant expects the costs of the plans to change next year

as follows. For individuals, the costs for the compre-

hensive, HMO standard, and HMO Plus plans will be

$683.91, $463.10, and $499.27, respectively. For fami-

lies, the costs for the comprehensive, HMO standard,

and HMO Plus plans will be $1699.48, $1217.45, and

$1273.08, respectively.

(a) Organize the information using two matrices and

where represents the health care plan costs for

this year and represents the health care plan costs

for next year. State what each entry of each matrix

represents.

(b) Compute and interpret the result.

(c) The employees receive monthly paychecks from

which the health care plan costs are deducted. Use

the matrices from part (a) to write matrices that show

how much will be deducted from each employees’

paycheck this year and next year.

(d) Suppose instead that the costs of the health care

plans increase by 4% next year. Write a matrix that

shows the new monthly payments.

EXPLORATION


77. Two matrices can be added only if they have the same

order.

78. Matrix multiplication is commutative.

THINK ABOUT IT In Exercises 79–86, let matrices and be of orders and respectively.Determine whether the matrices are of proper order to perform the operation(s). If so, give the order of the answer.

79. 80.

81. 82.

83. 84.

85. 86.

87. Consider matrices and below. Perform the

indicated operations and compare the results.

(a) Find and

(b) Find then add to the resulting matrix.

Find then add to the resulting matrix.

(c) Find and then add the two resulting matrices.

Find then multiply the resulting matrix by 2.

88. Use the following matrices to find and

What do your results tell you about matrix

multiplication, commutativity, and associativity?

89. THINK ABOUT IT If and are real numbers such

that and then However, if

and are nonzero matrices such that then

is not necessarily equal to Illustrate this using the

following matrices.

90. THINK ABOUT IT If and are real numbers such

that then or However, if and

are matrices such that it is not necessarily true

that or Illustrate this using the following

matrices.

91. Let and be unequal diagonal matrices of the same

order. (A diagonal matrix is a square matrix in which

each entry not on the main diagonal is zero.) Determine

the products for several pairs of such matrices. Make

a conjecture about a quick rule for such products.

92. Let and let

and

(a) Find and Identify any similarities with

and

(b) Find and identify

93. Find two matrices and such that AB � BA.BA

B2.

i 4.i 3,

i 2,A4.A3,A2,

B � �0

i

�i

0�.A � � i00

i�i � �1

AB

BA

B � � 1

�1

�1

1�A � �3

4

3

4�,

B � O.A � O

AB � O,

BAb � 0.a � 0ab � 0,

ba

C � �2

2

3

3�B � �1

1

0

0�,A � �0

0

1

1�,

B.

AAC � BC,C

B,A,a � b.ac � bc,c � 0

cb,a,

C � �3

0

0

1�B � �0

2

1

3�,A � �1

3

2

4�,

A�BC�.�AB�C,BA,AB,

A � B,

2B,2A

AB � C,

CA � B,

B � A.A � B

C � �5

2

2

�6�B � ��2

8

0

1�,A � �3

4

�1

7�,

CB,A,

�BC � D�AD�A � 3B�CB � DBC � D

BCAB

B � 3CA � 2C

2 2,3 2,2 3,2 3,D

C,B,A,

A � B

B

A

B,A

94. CAPSTONE Let matrices and be of orders

and respectively. Answer the following

questions and explain your reasoning.

(a) Is it possible that

(b) Is defined?

(c) Is defined? If so, is it possible that AB � BA?AB

A � B

A � B?

2 2,3 2

BA

Section 8.3 The Inverse of a Square Matrix 599

8.3 THE INVERSE OF A SQUARE MATRIX


• Verify that two matrices are inverses of each other.

• Use Gauss-Jordan elimination to find the inverses of matrices.

• Use a formula to find the inverses of matrices.

• Use inverse matrices to solve systems of linear equations.


You can use inverse matrices to model and solve real-life problems.For instance, in Exercise 75 on page607, an inverse matrix is used to finda quadratic model for the enrollmentprojections for public universities inthe United States.

2 2

Alberto L. Pomares/istockphoto.com

The Inverse of a Matrix

This section further develops the algebra of matrices. To begin, consider the real number

equation To solve this equation for multiply each side of the equation by

(provided that ).

The number is called the multiplicative inverse of because The definition

of the multiplicative inverse of a matrix is similar.

The Inverse of a Matrix

Show that is the inverse of where

and

Solution

To show that is the inverse of show that as follows.

As you can see, This is an example of a square matrix that has an

inverse. Note that not all square matrices have inverses.

Now try Exercise 5.

Recall that it is not always true that even if both products are defined.

However, if and are both square matrices and it can be shown that

So, in Example 1, you need only to check that AB � I2.BA � In .

AB � In ,BA

AB � BA,

AB � I � BA.

BA � �1

1

�2

�1��1

�1

2

1� � ��1 � 2

�1 � 1

2 � 2

2 � 1� � �1

0

0

1�

AB � ��1

�1

2

1� �1

1

�2

�1� � ��1 � 2

�1 � 1

2 � 2

2 � 1� � �1

0

0

1�AB � I � BA,A,B

B � �1

1

�2

�1�.A � ��1

�1

2

1�A,B

Example 1

a�1a � 1.aa�1

x � a�1b

�1�x � a�1b

�a�1a�x � a�1b

ax � b

a � 0

a�1x,ax � b.

Definition of the Inverse of a Square Matrix

Let be an matrix and let be the identity matrix. If there exists a

matrix such that

then is called the inverse of The symbol is read “ inverse.”AA�1A.A�1

AA�1 � In � A�1A

A�1

n nInn nA

Finding Inverse Matrices

If a matrix has an inverse, is called invertible (or nonsingular); otherwise, is

called singular. A nonsquare matrix cannot have an inverse. To see this, note that if

is of order and is of order (where ), the products and are

of different orders and so cannot be equal to each other. Not all square matrices have

inverses (see the matrix at the bottom of page 602). If, however, a matrix does have an

inverse, that inverse is unique. Example 2 shows how to use a system of equations to

find the inverse of a matrix.

Finding the Inverse of a Matrix

Find the inverse of

Solution

To find the inverse of try to solve the matrix equation for

A X I

Equating corresponding entries, you obtain two systems of linear equations.

Linear system with two variables, and

Linear system with two variables, and

Solve the first system using elementary row operations to determine that and

From the second system you can determine that and

Therefore, the inverse of is

You can use matrix multiplication to check this result.

Check

�

�


A�1A � ��3

1

�4

1��1

�1

4

�3� � �1

0

0

1�

AA�1 � � 1

�1

4

�3��3

1

�4

1� � �1

0

0

1�

� ��3

1

�4

1�.

X � A�1

A

x22 � 1.x12 � �4x21 � 1.

x11 � �3

x22.x12� x12 � 4x22 � 0

�x12 � 3x22 � 1

x21.x11� x11 � 4x21 � 1

�x11 � 3x21 � 0

� x11 � 4x21

�x11 � 3x21

x12 � 4x22

�x12 � 3x22� � �1

0

0

1�

� 1

�1

4

�3��x11

x21

x12

x22� � �1

0

0

1�

X.AX � IA,

A � � 1

�1

4

�3�.

Example 2

BAABm � nn mBm n

A

AAA



In Example 2, note that the two systems of linear equations have the same

coefficient matrix Rather than solve the two systems represented by

and

separately, you can solve them simultaneously by adjoining the identity matrix to the

coefficient matrix to obtain

A I

This “doubly augmented” matrix can be represented as By applying

Gauss-Jordan elimination to this matrix, you can solve both systems with a single

elimination process.

So, from the “doubly augmented” matrix you obtain the matrix

A I I

This procedure (or algorithm) works for any square matrix that has an inverse.

�1

0

0

1

.

..

.

..�3

1

�4

1�� 1

�1

4

�3

.

..

.

..1

0

0

1�A�1

�I A�1�.�A I�,

�4R2 � R1 → � 1

0

0

1

.

..

.

..�3

1

�4

1�R1 � R2 →

� 1

0

4

1

.

..

.

..1

1

0

1�

� 1

�1

4

�3

.

..

.

..1

0

0

1�

�A I�.

� 1

�1

4

�3

.

..

.

..1

0

0

1�.

� 1

�1

4

�3

.

..

.

..0

1�

� 1

�1

4

�3

.

..

.

..1

0�A.

Finding an Inverse Matrix

Let be a square matrix of order

1. Write the matrix that consists of the given matrix on the left and the

identity matrix on the right to obtain

2. If possible, row reduce to using elementary row operations on the entire

matrix The result will be the matrix If this is not possible,

is not invertible.

3. Check your work by multiplying to see that AA�1 � I � A�1A.

A

�I A�1�.�A I�.IA

�A I�.In n

An 2n

n.A

TECHNOLOGY

Most graphing utilities can findthe inverse of a square matrix.To do so, you may have to usethe inverse key .Consult theuser’s guide for your graphingutility for specific keystrokes.

x�1

Finding the Inverse of a Matrix

Find the inverse of

Solution

Begin by adjoining the identity matrix to to form the matrix

Use elementary row operations to obtain the form as follows.

So, the matrix is invertible and its inverse is

Confirm this result by multiplying and to obtain as follows.

Check


The process shown in Example 3 applies to any matrix When using this

algorithm, if the matrix does not reduce to the identity matrix, then does not have

an inverse. For instance, the following matrix has no inverse.

To confirm that matrix above has no inverse, adjoin the identity matrix to to form

and perform elementary row operations on the matrix. After doing so, you will

see that it is impossible to obtain the identity matrix on the left. Therefore, is not

invertible.

AI

�A I�AA

A � �1

3

�2

2

�1

3

0

2

�2�

AA

A.n n

AA�1 � �1

1

6

�1

0

�2

0

�1

�3� ��2

�3

�2

�3

�3

�4

1

1

1� � �

1

0

0

0

1

0

0

0

1� � I

I,A�1A

A�1 � ��2

�3

�2

�3

�3

�4

1

1

1�.

A

R3 � R1 →

R3 � R2 → �1

0

0

0

1

0

0

0

1

.

..

.

..

.

..

�2

�3

�2

�3

�3

�4

1

1

1� � �I .

.. A�1�

R2 � R1 →

�4R2 � R3 →�1

0

0

0

1

0

�1

�1

1

.

..

.

..

.

..

0

�1

�2

1

1

�4

0

0

1�

�R1 � R2 →

�6R1 � R3 →�1

0

0

�1

1

4

0

�1

�3

.

..

.

..

.

..

1

�1

�6

0

1

0

0

0

1�

�I A�1�,

�A... I� � �

1

1

6

�1

0

�2

0

�1

�3

.

..

.

..

.

..

1

0

0

0

1

0

0

0

1�.

A

A � �1

1

6

�1

0

�2

0

�1

�3�.

Example 3


WARNING / CAUTION

Be sure to check your solution

because it is easy to make

algebraic errors when using

elementary row operations.


The Inverse of a 2 2 Matrix

Using Gauss-Jordan elimination to find the inverse of a matrix works well (even as a

computer technique) for matrices of order or greater. For matrices, however,

many people prefer to use a formula for the inverse rather than Gauss-Jordan

elimination. This simple formula, which works only for matrices, is explained as

follows. If is a matrix given by

then is invertible if and only if Moreover, if the inverse

is given by

Formula for inverse of matrix A

The denominator is called the determinant of the matrix You will

study determinants in the next section.

Finding the Inverse of a 2 2 Matrix

If possible, find the inverse of each matrix.

a.

b.

Solution

a. For the matrix apply the formula for the inverse of a matrix to obtain

Because this quantity is not zero, the inverse is formed by interchanging the entries

on the main diagonal, changing the signs of the other two entries, and multiplying

by the scalar as follows.

Substitute for and the determinant.

Multiply by the scalar

b. For the matrix you have

which means that is not invertible.


B

� 0

ad � bc � �3��2� � ��1��6�

B,

14.� �

12

12

14

34�

d,c,b,a,A�1 �14�2

2

1

3�

14,

� 4.

ad � bc � �3��2� � ��1��2�

2 2A,

B � � 3

�6

�1

2�

A � � 3

�2

�1

2�

Example 4

A.2 2ad � bc

A�1 �1

ad � bc �d

�c

�b

a�.

ad � bc � 0,ad � bc � 0.A

A � �acb

d�2 2A

2 2

2 23 3


Systems of Linear Equations

You know that a system of linear equations can have exactly one solution, infinitely

many solutions, or no solution. If the coefficient matrix of a square system (a system

that has the same number of equations as variables) is invertible, the system has a

unique solution, which is defined as follows.

Solving a System Using an Inverse Matrix

You are going to invest $10,000 in AAA-rated bonds, AA-rated bonds, and

B-rated bonds and want an annual return of $730. The average yields are 6% on AAA

bonds, 7.5% on AA bonds, and 9.5% on B bonds. You will invest twice as much in AAA

bonds as in B bonds. Your investment can be represented as

where and represent the amounts invested in AAA, AA, and B bonds, respectively.

Use an inverse matrix to solve the system.

Solution

Begin by writing the system in the matrix form

Then, use Gauss-Jordan elimination to find

Finally, multiply by on the left to obtain the solution.

The solution of the system is and So, you will invest

$4000 in AAA bonds, $4000 in AA bonds, and $2000 in B bonds.


z � 2000.y � 4000,x � 4000,

� �4000

4000

2000��

15

�21.5

7.5

�200

300

�100

�2

3.5

�1.5� �

10,000

730

0�

X � A�1B

A�1B

A�1 � �15

�21.5

7.5

�200

300

�100

�2

3.5

�1.5�

A�1.

�x

y

z� � �

10,000

730

0��

1

0.06

1

1

0.075

0

1

0.095

�2�

AX � B.

zy,x,

� x

0.06x

x

�

�

y

0.075y

�

�

�

z �

0.095z �

2z �

10,000

730

0

Example 5

A

A System of Equations with a Unique Solution

If is an invertible matrix, the system of linear equations represented by

has a unique solution given by

X � A�1B.

AX � B

A

TECHNOLOGY

To solve a system of equationswith a graphing utility, enter thematrices and in the matrix editor. Then, using the inversekey, solve for

The screen will display thesolution, matrix X.

ENTERBA

X.

BA

x�1


EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises.8.3VOCABULARY: Fill in the blanks.

1. In a ________ matrix, the number of rows equals the number of columns.

2. If there exists an matrix such that then is called the ________ of

3. If a matrix has an inverse, it is called invertible or ________; if it does not have an inverse, it is called ________.

4. If is an invertible matrix, the system of linear equations represented by has a unique solution given by

________.


X �

AX � BA

A

A.A�1AA�1 � In � A�1A,A�1n n

In Exercises 5–12, show that is the inverse of

5.

6.

7.

8.

9.

10.

11.

12.

In Exercises 13–24, find the inverse of the matrix (if it exists).

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

In Exercises 25–34, use the matrix capabilities of a graphingutility to find the inverse of the matrix (if it exists).

25. 26.

27. 28.

29. 30.

31. 32.

33. 34.

In Exercises 35–40, use the formula on page 603 to find theinverse of the matrix (if it exists).

35. 36.

37. 38. ��12

5

3

�2��4

2

�6

3�� 1

�3

�2

2�� 2

�1

3

5�2 2

�1

3

2

�1

�2

�5

�5

4

�1

�2

�2

4

�2

�3

�5

11��

�1

0

2

0

0

2

0

�1

1

0

�1

0

0

�1

0

1�

�0.6

0.7

1

0

�1

0

�0.3

0.2

�0.9��

0.1

�0.3

0.5

0.2

0.2

0.4

0.3

0.2

0.4�

��56

0

1

13

23

�12

116

2

�52

��12

1

0

34

0

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14

�3212

��

3

2

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2

2

4

2

2

3��

1

3

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1

1

0

2

0

3�

�10

�5

3

5

1

2

�7

4

�2��

1

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2

7

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6

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0

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�5�

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5

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5��

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2

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0

0

5

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0

7�

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3

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2

7

�4

2

9

�7��

1

3

3

1

5

6

1

4

5�

� 4

�3

�1

1��3

4

1

2��7

4

33

�19��1

2

�2

�3��1

3

2

7��2

0

0

3�

B �1

3��3

�3

0

�3

1

�1

1

�2

1

2

1

1

�3

�3

0

0�

A � ��1

1

�1

0

1

�1

1

�1

0

1

2

1

�1

0

0

1�,

A � ��2

1

0

2

�1

1

3

0

4�, B �

1

3��4

�4

1

�5

�8

2

3

3

0�

B � ��12

14

�14

1

�1

1

32

�114

74

�A � ��4

�1

0

1

2

�1

5

4

�1�,

A � �2

�1

0

�17

11

3

11

�7

�2�, B � �

1

2

3

1

4

6

2

�3

�5�

B � �35

�25

15

15�A � �1

2

�1

3�,

B � ��232

1

�12�A � �1

3

2

4�,

B � �2

1

1

1�A � � 1

�1

�1

2�,

B � � 3

�5

�1

2�A � �2

5

1

3�,

A.B


39. 40.

In Exercises 41–44, use the inverse matrix found in Exercise15 to solve the system of linear equations.

41. 42.

43. 44.

In Exercises 45 and 46, use the inverse matrix found inExercise 19 to solve the system of linear equations.

45. 46.

In Exercises 47 and 48, use the inverse matrix found inExercise 34 to solve the system of linear equations.

47.

48.

In Exercises 49 and 50, use a graphing utility to solve the system of linear equations using an inverse matrix.

49.

50.

In Exercises 51–58, use an inverse matrix to solve (if possible)the system of linear equations.

51. 52.

53. 54.

55. 56.

57. 58.

In Exercises 59–62, use the matrix capabilities of a graphingutility to solve (if possible) the system of linear equations.

59. 60.

61.

62.

In Exercises 63 and 64, show that the matrix is invertible andfind its inverse.

63. 64.

INVESTMENT PORTFOLIO In Exercises 65–68, consider aperson who invests in AAA-rated bonds, A-rated bonds, and B-rated bonds. The average yields are 6.5% on AAA bonds, 7%on A bonds, and 9% on B bonds. The person invests twice asmuch in B bonds as in A bonds. Let and represent theamounts invested in AAA, A, and B bonds, respectively.

Use the inverse of the coefficient matrix of this system to findthe amount invested in each type of bond.

Total Investment Annual Return

65. $10,000 $705

66. $10,000 $760

67. $12,000 $835

68. $500,000 $38,000

PRODUCTION In Exercises 69–72, a small home businesscreates muffins, bones, and cookies for dogs. In addition toother ingredients, each muffin requires 2 units of beef, 3 unitsof chicken, and 2 units of liver. Each bone requires 1 unit ofbeef, 1 unit of chicken, and 1 unit of liver. Each cookie requires2 units of beef, 1 unit of chicken, and 1.5 units of liver. Find thenumbers of muffins, bones, and cookies that the company cancreate with the given amounts of ingredients.

� x

0.065x

1

1

y 1

0.07y 1

2y !

z "

0.09z "

z "

!total investment"!annual return"0

zy,x,

A � �sec "

tan "

tan "

sec "�A � � sin "

�cos "

cos "

sin "�

��8x � 7y � 10z �

12x � 3y � 5z �

15x � 9y � 2z �

�151

86

187

�3x � 2y � z �

�4x � y � 3z �

x � 5y � z �

�29

37

�24

�2x � 3y � 5z � 4

3x � 5y � 9z � 7

5x � 9y � 17z � 13�

5x � 3y � 2z �

2x � 2y � 3z �

x � 7y � 8z �

2

3

�4

�4x � 2y � 3z �

2x � 2y � 5z �

8x � 5y � 2z �

�2

16

4�

4x � y � z �

2x � 2y � 3z �

5x � 2y � 6z �

�5

10

1

�56 x �43 x �

y � �2072 y � �51��

14 x �

38 y � �2

32 x �

34 y � �12

�0.2x �

�x �

0.6y �

1.4y �

2.4

�8.8��0.4x �

2x �

0.8y �

4y �

1.6

5

�18x � 12y � 13

30x � 24y � 23�3x � 4y �

5x � 3y �

�2

4

x1

2x1

x1

2x1

3x1

�

�

�

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x2

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3x1 � 5x2 � 2x3 � 3x4 �

2x1 � 5x2 � 2x3 � 5x4 �

�x1 � 4x2 � 4x3 � 11x4 �

1

�2

0

�3

�x1 � 2x2 � x3 � 2x4 �

3x1 � 5x2 � 2x3 � 3x4 �

2x1 � 5x2 � 2x3 � 5x4 �

�x1 � 4x2 � 4x3 � 11x4 �

0

1

�1

2

�x � y � z �

3x � 5y � 4z �

3x � 6y � 5z �

�1

2

0�x � y � z � 0

3x � 5y � 4z � 5

3x � 6y � 5z � 2

� x � 2y �

2x � 3y �

1

�2� x � 2y � 4

2x � 3y � 2

� x � 2y � 0

2x � 3y � 3� x � 2y � 5

2x � 3y � 10

��14

53

94

89��

72

15

�34

45�


69. 700 units of beef 70. 525 units of beef

500 units of chicken 480 units of chicken

600 units of liver 500 units of liver

71. 800 units of beef 72. 1000 units of beef

750 units of chicken 950 units of chicken

725 units of liver 900 units of liver

73. COFFEE A coffee manufacturer sells a 10-pound

package that contains three flavors of coffee for $26.

French vanilla coffee costs $2 per pound, hazelnut fla-

vored coffee costs $2.50 per pound, and Swiss chocolate

flavored coffee costs $3 per pound. The package contains

the same amount of hazelnut as Swiss chocolate. Let

represent the number of pounds of French vanilla, rep-

resent the number of pounds of hazelnut, and represent

the number of pounds of Swiss chocolate.

(a) Write a system of linear equations that represents

the situation.

(b) Write a matrix equation that corresponds to your

system.

(c) Solve your system of linear equations using an

inverse matrix. Find the number of pounds of each

flavor of coffee in the 10-pound package.

74. FLOWERS A florist is creating 10 centerpieces for the

tables at a wedding reception. Roses cost $2.50 each,

lilies cost $4 each, and irises cost $2 each. The customer

has a budget of $300 allocated for the centerpieces and

wants each centerpiece to contain 12 flowers, with twice

as many roses as the number of irises and lilies combined.

(a) Write a system of linear equations that represents

the situation.

(b) Write a matrix equation that corresponds to your system.

(c) Solve your system of linear equations using an inverse

matrix. Find the number of flowers of each type that

the florist can use to create the 10 centerpieces.

75. ENROLLMENT The table shows the enrollment

projections (in millions) for public universities in the

United States for the years 2010 through 2012. (Source:

U.S. National Center for Education Statistics, Digest of

Education Statistics)

(a) The data can be modeled by the quadratic function

Create a system of linear equations

for the data. Let represent the year, with

corresponding to 2010.

(b) Use the matrix capabilities of a graphing utility to find

the inverse matrix to solve the system from part (a)

and find the least squares regression parabola

(c) Use the graphing utility to graph the parabola with the

data.

(d) Do you believe the model is a reasonable predictor of

future enrollments? Explain.

EXPLORATION


77. Multiplication of an invertible matrix and its inverse is

commutative.

78. If you multiply two square matrices and obtain the

identity matrix, you can assume that the matrices are

inverses of one another.

79. WRITING Explain how to determine whether the

inverse of a matrix exists. If so, explain how to

find the inverse.

80. WRITING Explain in your own words how to write a

system of three linear equations in three variables as a

matrix equation, as well as how to solve the

system using an inverse matrix.

81. Consider matrices of the form

(a) Write a matrix and a matrix in the form

of Find the inverse of each.

(b) Use the result of part (a) to make a conjecture about

the inverses of matrices in the form of

PROJECT: VIEWING TELEVISION To work an extended

application analyzing the average amounts of time spent

viewing television in the United States, visit this text’s

website at academic.cengage.com. (Data Source: The

Nielsen Company)

A.

A.

3 32 2

A � �a11

0

0

0

0

a22

0

0

0

0

a33

0

0

0

0

0

. . .

. . .

. . .

. . .

. . .

0

0

0

ann

�.

AX � B,

2 2

y � at2 � bt � c.

t � 10t

y � at2 � bt � c.

s

h

f

76. CAPSTONE If is a matrix

then is invertible if and only if If

verify that the inverse is

A�1 �1

ad � bc �d

�c

�b

a�.

ad � bc � 0,

ad � bc � 0.A

A � �acb

d�,2 2A

Year Enrollment projections

2010

2011

2012

13.89

14.04

14.20


8.4 THE DETERMINANT OF A SQUARE MATRIX


• Find the determinants of matrices.

• Find minors and cofactors of square matrices.

• Find the determinants of squarematrices.


Determinants are often used in otherbranches of mathematics. For instance,Exercises 85–90 on page 615 showsome types of determinants that areuseful when changes in variables aremade in calculus.

2 2The Determinant of a 2 2 Matrix

Every square matrix can be associated with a real number called its determinant.

Determinants have many uses, and several will be discussed in this and the next section.

Historically, the use of determinants arose from special number patterns that occur

when systems of linear equations are solved. For instance, the system

has a solution

and

provided that Note that the denominators of the two fractions are the

same. This denominator is called the determinant of the coefficient matrix of the system.

Coefficient Matrix Determinant

The determinant of the matrix can also be denoted by vertical bars on both sides of

the matrix, as indicated in the following definition.

In this text, and are used interchangeably to represent the determinant of

Although vertical bars are also used to denote the absolute value of a real number, the

context will show which use is intended.

A convenient method for remembering the formula for the determinant of a

matrix is shown in the following diagram.

Note that the determinant is the difference of the products of the two diagonals of the

matrix.

det�A� � a1

a2

b1

b2 � a1b2 � a2b1

2 2

A.Adet�A�

A

det�A� � a1b2 � a2b1A � �a1

a2

b1

b2�

a1b2 � a2b1 � 0.

y �a1c2 � a2c1

a1b2 � a2b1

x �c1b2 � c2b1

a1b2 � a2b1

�a1x � b1y � c1

a2x � b2y � c2

Definition of the Determinant of a 2 2 Matrix

The determinant of the matrix

is given by

det�A� � A � a1

a2

b1

b2 � a1b2 � a2b1.

A � �a1

a2

b1

b2�

Section 8.4 The Determinant of a Square Matrix 609

The Determinant of a 2 2 Matrix

Find the determinant of each matrix.

a.

b.

c.

Solution

a.

b.

c.

Now try Exercise 9.

Notice in Example 1 that the determinant of a matrix can be positive, zero, or

negative.

The determinant of a matrix of order is defined simply as the entry of the

matrix. For instance, if then det�A� � �2.A � ��2�,1 1

� 0 � 3 � �3

� 0�4� � 2�32�

det�C� � 02 32

4� 4 � 4 � 0

� 2�2� � 4�1�

det�B� � 24 1

2� 7� 4 � 3

� 2�2� � 1��3�

det�A� � 21 �3

2

C � �0

2

32

4�B � �2

4

1

2�

A � �2

1

�3

2�

Example 1

TECHNOLOGY

Most graphing utilities can evaluate the determinant of a matrix. For instance, youcan evaluate the determinant of

by entering the matrix as and then choosing the determinant feature. The resultshould be 7, as in Example 1(a). Try evaluating the determinants of other matrices.Consult the user’s guide for your graphing utility for specific keystrokes.

[A]A " [2

1

!3

2]


Minors and Cofactors

To define the determinant of a square matrix of order or higher, it is convenient

to introduce the concepts of minors and cofactors.

In the sign pattern for cofactors at the left, notice that odd positions (where is

odd) have negative signs and even positions (where is even) have positive signs.

Finding the Minors and Cofactors of a Matrix

Find all the minors and cofactors of

Solution

To find the minor delete the first row and first column of and evaluate the

determinant of the resulting matrix.

Similarly, to find delete the first row and second column.

Continuing this pattern, you obtain the minors.

Now, to find the cofactors, combine these minors with the checkerboard pattern of signs

for a matrix shown at the upper left.


C33 � �6C32 � 3C31 � 5

C23 � 8C22 � �4C21 � �2

C13 � 4C12 � 5C11 � �1

3 3

M33 � �6M32 � �3M31 � 5

M23 � �8M22 � �4M21 � 2

M13 � 4M12 � �5M11 � �1

M12 � 34 2

1 � 3�1� � 4�2� � �5�0

3

4

2

�1

0

1

2

1�,

M12,

M11 � �1

0

2

1 � �1�1� � 0�2� � �1�0

3

4

2

�1

0

1

2

1�,

AM11,

A � �0

3

4

2

�1

0

1

2

1�.

Example 2

i � j

i � j

3 3

Minors and Cofactors of a Square Matrix

If is a square matrix, the minor of the entry is the determinant of the

matrix obtained by deleting the th row and th column of The cofactor

of the entry is

Ci j � ��1�i� jMi j.

ai j

Ci jA.ji

ai jMi jA

Sign Pattern for Cofactors

matrix

matrix

matrixn n

��

�

�

�

�

.

..

�

�

�

�

�

.

..

�

�

�

�

�

.

..

�

�

�

�

�

.

..

�

�

�

�

�

.

..

. . .

. . .

. . .

. . .

. . .�4 4

��

�

�

�

�

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�

�

�

�

�

�

�

�

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�3 3

��

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��


The Determinant of a Square Matrix

The definition below is called inductive because it uses determinants of matrices of

order to define determinants of matrices of order

Try checking that for a matrix

this definition of the determinant yields as previously defined.

The Determinant of a Matrix of Order 3 3

Find the determinant of

Solution

Note that this is the same matrix that was in Example 2. There you found the cofactors

of the entries in the first row to be

and

So, by the definition of a determinant, you have

First-row expansion


In Example 3, the determinant was found by expanding by the cofactors in the first

row. You could have used any row or column. For instance, you could have expanded

along the second row to obtain

Second-row expansion

� 14.

� 3��2� � ��1��4� � 2�8�A � a21C21 � a22C22 � a23C23

� 14.

� 0��1� � 2�5� � 1�4�A � a11C11 � a12C12 � a13C13

C13 � 4.C12 � 5,C11 � �1,

A � �0

3

4

2

�1

0

1

2

1�.

Example 3

A � a1b2 � a2b1,

A � �a1

a2

b1

b2�

2 2

n.n � 1

Determinant of a Square Matrix

If is a square matrix (of order or greater), the determinant of is the

sum of the entries in any row (or column) of multiplied by their respective

cofactors. For instance, expanding along the first row yields

Applying this definition to find a determinant is called expanding by cofactors.

A � a11C11 � a12C12 �. . . � a1nC1n.

A

A2 2A

When expanding by cofactors, you do not need to find cofactors of zero entries,

because zero times its cofactor is zero.

So, the row (or column) containing the most zeros is usually the best choice for expansion

by cofactors. This is demonstrated in the next example.

The Determinant of a Matrix of Order 4 4

Find the determinant of

Solution

After inspecting this matrix, you can see that three of the entries in the third column are

zeros. So, you can eliminate some of the work in the expansion by using the third

column.

Because and have zero coefficients, you need only find the cofactor

To do this, delete the first row and third column of and evaluate the determinant of

the resulting matrix.

Delete 1st row and 3rd column.

Simplify.

Expanding by cofactors in the second row yields

So, you obtain


Try using a graphing utility to confirm the result of Example 4.

� 15.

� 3�5�A � 3C13

� 5.

� 0 � 2�1��8� � 3��1��7�

C13 � 0��1�314 2

2 � 2��1�4�1

3

2

2 � 3��1�5�1

3

1

4

� �1

0

3

1

2

4

2

3

2C13 � ��1�1�3�1

0

3

1

2

4

2

3

2A

C13.C43C23, C33,

A � 3�C13� � 0�C23� � 0�C33� � 0�C43�

A � �1

�1

0

3

�2

1

2

4

3

0

0

0

0

2

3

2�.

Example 4

aijCij � �0�Cij � 0



EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises.8.4VOCABULARY: Fill in the blanks.

1. Both and represent the ________ of the matrix

2. The ________ of the entry is the determinant of the matrix obtained by deleting the th row and th

column of the square matrix

3. The ________ of the entry of the square matrix is given by

4. The method of finding the determinant of a matrix of order or greater is called ________ by ________.


2 2

��1�i� jMij.AaijCij

A.

jiaijMij

A.Adet�A�

In Exercises 5–20, find the determinant of the matrix.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

In Exercises 21–24, use the matrix capabilities of a graphingutility to find the determinant of the matrix.

21. 22.

23. 24.

In Exercises 25–32, find all (a) minors and (b) cofactors of thematrix.

25. 26.

27. 28.

29. 30.

31. 32.

In Exercises 33–38, find the determinant of the matrix by the method of expansion by cofactors. Expand using the indicated row or column.

33. 34.

(a) Row 1 (a) Row 2

(b) Column 2 (b) Column 3

35. 36.

(a) Row 2 (a) Row 3


37. 38.

(a) Row 2 (a) Row 3


In Exercises 39–54, find the determinant of the matrix.Expand by cofactors on the row or column that appears tomake the computations easiest.

39. 40.

41. 42.

43. 44. �1

�1

4

0

�1

11

0

0

5��

�1

0

0

8

3

0

�3

�6

3�

�1

3

�2

1

1

0

2

0

3��

6

0

4

3

0

�6

�7

0

3�

��2

1

0

2

�1

1

3

0

4��

2

4

4

�1

2

2

0

1

1�

�10

4

0

1

8

0

3

0

3

5

2

�3

�7

�6

7

2��

6

4

�1

8

0

13

0

6

�3

6

7

0

5

�8

4

2�

�10

30

0

�5

0

10

5

10

1��

5

0

1

0

12

6

�3

4

3�

��3

6

4

4

3

�7

2

1

�8��

�3

4

2

2

5

�3

1

6

1�

��2

7

6

9

�6

7

4

0

�6��

�4

7

1

6

�2

0

3

8

�5�

�1

3

4

�1

2

�6

0

5

4��

4

�3

1

0

2

�1

2

1

1�

��6

7

5

�2�� 3

�2

1

�4��0

3

10

�4��4

3

5

�6�

�0.1

7.5

0.3

0.1

6.2

0.6

�4.3

0.7

�1.2��

0.9

�0.1

�2.2

0.7

0.3

4.2

0

1.3

6.1�

�0.1

�0.3

0.5

0.2

0.2

0.4

0.3

0.2

0.4��

0.3

0.2

�0.4

0.2

0.2

0.4

0.2

0.2

0.3�

�23

�1

43

�13��

12

�6

1313�

� 0

�3

6

2��712

6

3�� 4

�2

7

5��3

�6

�2

�1�� 2

�6

�3

9��2

0

6

3��4

0

�3

0��7

3

0

0��3

4

�3

�8�� 6

�5

2

3��9

6

0

2��8

2

4

3��10��4�


45. 46.

47. 48.

49. 50.

51. 52.

53.

54.

In Exercises 55–62, use the matrix capabilities of a graphingutility to evaluate the determinant.

55. 56.

57. 58.

59. 60.

61.

62.

In Exercises 63–70, find (a) (b) (c) and (d)

63.

64.

65.

66.

67.

68.

69.

70.

In Exercises 71–76, evaluate the determinant(s) to verify theequation.

71. 72.

73.

74.

75.

76.

In Exercises 77–84, solve for

77. 78.

79. 80.

81. 82.

83. 84. x � 4

7

�2

x � 5 � 0x � 3

1

2

x � 2 � 0

x � 2

�3

�1

x � 0x � 1

3

2

x � 2 � 0

x � 1

�1

2

x � 4x2 1

x � 2 � �1

x

�1

4

x � 20x1 2

x � 2

x.

a � b

a

a

a

a � b

a

a

a

a � b � b2�3a � b�

111 x

y

z

x 2

y 2

z 2 � �y � x��z � x��z � y�

wcw x

cx � 0

wy x

z � wy x � cw

z � cy wy cx

cz � cwy x

zwy x

z � � yw z

x

B � �2

0

3

�1

1

�2

4

3

1�A � �2

1

3

0

�1

1

1

2

0�,

B � ��1

0

0

0

2

0

0

0

3�A � ��1

1

0

2

0

1

1

1

0�,

B � ��3

0

�2

0

2

�1

1

�1

1�A � �

3

�1

�2

2

�3

0

0

4

1�,

B � �3

1

3

�2

�1

1

0

2

1�A � �

0

�3

0

1

�2

4

2

1

1�,

B � �0

1

6

�2�A � �5

3

4

�1�,

B � ��1

�2

1

2�A � �4

3

0

�2�,

B � �1

0

2

�1�A � ��2

4

1

�2�,

B � �2

0

0

�1�A � ��1

0

0

3�,

AB.AB,B,A,

�2

0

0

0

0

0

3

0

0

0

0

0

�1

0

0

0

0

0

2

0

0

0

0

0

�4 3

�1

5

4

1

�2

0

�1

7

2

4

2

0

�8

3

3

1

3

0

0

1

0

2

0

2 0

8

�4

�7

�3

1

6

0

8

�1

0

0

2

6

9

141220 �1

6

0

2

8

0

2

8

4

�4

6

0 3

�2

12

0

5

5

0

0

7 7

�2

�6

0

5

2

�14

4

12 5

9

�8

�8

7

7

0

4

1308 8

�5

1

�7

4

6

�5

0

0

0

0

2

1

0

0

0

0

4

2

3

0

0

3

6

4

0

�2

2

3

1

2

��

3

�2

1

6

3

2

0

0

0

0

4

1

0

2

5

�1

3

4

�1

1

5

2

0

0

0

��

1

�5

0

3

4

6

0

�2

3

2

0

1

2

1

0

5��

5

4

0

0

3

6

2

1

0

4

�3

�2

6

12

4

2�

�3

�2

1

0

6

0

1

3

�5

6

2

�1

4

0

2

�1��

2

2

1

3

6

7

5

7

6

3

0

0

2

6

1

7�

��3

7

1

0

11

2

0

0

2��

2

0

0

4

3

0

6

1

�5�

�2

1

1

�1

4

0

3

4

2��

1

3

�1

4

2

4

�2

0

3�


In Exercises 85–90, evaluate the determinant in which theentries are functions. Determinants of this type occur whenchanges of variables are made in calculus.

85. 86.

87. 88.

89. 90.

EXPLORATION


91. If a square matrix has an entire row of zeros, the

determinant will always be zero.

92. If two columns of a square matrix are the same, the

determinant of the matrix will be zero.

93. Find square matrices and to demonstrate that

94. Consider square matrices in which the entries are

consecutive integers. An example of such a matrix is

(a) Use a graphing utility to evaluate the determinants

of four matrices of this type. Make a conjecture

based on the results.

(b) Verify your conjecture.

95. WRITING Write a brief paragraph explaining the

difference between a square matrix and its determinant.

96. THINK ABOUT IT If is a matrix of order such

that is it possible to find Explain.

PROPERTIES OF DETERMINANTS In Exercises 97–99, a property of determinants is given ( and are square matrices). State how the property has been applied to thegiven determinants and use a graphing utility to verify theresults.

97. If is obtained from by interchanging two rows of

or interchanging two columns of then

(a)

(b)

98. If is obtained from by adding a multiple of a row

of to another row of or by adding a multiple of a

column of to another column of then

(a)

(b)

99. If is obtained from by multiplying a row by a

nonzero constant or by multiplying a column by a

nonzero constant then

(a)

(b)

In Exercises 101–104, evaluate the determinant.

101. 102.

103. 104.

105. CONJECTURE A triangular matrix is a square

matrix with all zero entries either below or above its

main diagonal. A square matrix is upper triangular if

it has all zero entries below its main diagonal and is

lower triangular if it has all zero entries above its

main diagonal. A matrix that is both upper and lower

triangular is called diagonal. That is, a diagonal

matrix is a square matrix in which all entries above

and below the main diagonal are zero. In Exercises

101–104, you evaluated the determinants of triangular

matrices. Make a conjecture based on your results.

106. Use the matrix capabilities of a graphing utility to find the

determinant of What message appears on the screen?

Why does the graphing utility display this message?

A � �1

�1

3

2

0

�2�

A.

1

�4

5

0

�1

1

0

0

5�1

0

0

2

3

0

�5

4

3�2

0

0

0

0

2

0

0

0

0

1

0

0

0

0

3100 0

5

0

0

0

2

137 8

�12

4

�3

6

9 � 12137 2

�3

1

�1

2

352 10

�3 � 512 2

�3B � cA.c,

c

AB

527 4

�3

6

2

4

3 � 127 10

�3

6

�6

4

315 �3

2 � 10 �3

17B � A.A,A

AA

AB

1

�2

1

3

2

6

4

0

2 � � 1

�2

1

6

2

3

2

0

4 1

�7

6

3

2

1

4

�5

2 � � 1

�7

6

4

�5

2

3

2

1B��A.A,

AAB

BA

2A?A � 5,

3 3A

�4

7

10

5

8

11

6

9

12�.

A � B � A � B.BA

x1 x ln x

1 � ln x x1 ln x

1�x e�x�e�x

xe�x

�1 � x�e�x e2x

2e2x

e3x

3e3x3x 2

1

�3y 2

1 4u

�1

�1

2v

100. CAPSTONE If is an matrix, explain how to

find the following.

(a) The minor of the entry

(b) The cofactor of the entry

(c) The determinant of A

aijCij

aijMij

n nA

Cramer’s Rule

So far, you have studied three methods for solving a system of linear equations: substitution,

elimination with equations, and elimination with matrices. In this section, you will

study one more method, Cramer’s Rule, named after Gabriel Cramer (1704–1752).

This rule uses determinants to write the solution of a system of linear equations. To see

how Cramer’s Rule works, take another look at the solution described at the beginning

of Section 8.4. There, it was pointed out that the system

has a solution

and

provided that Each numerator and denominator in this solution can

be expressed as a determinant, as follows.

Relative to the original system, the denominator for and is simply the determinant

of the coefficient matrix of the system. This determinant is denoted by The numerators

for and are denoted by and respectively. They are formed by using the

column of constants as replacements for the coefficients of and as follows.

Coefficient

Matrix D

For example, given the system

the coefficient matrix, D, and are as follows.

Coefficient

Matrix D

2

�4

3

838 �5

3 2

�4

�5

3� 2

�4

�5

3�DyDx

DyDx,

� 2x � 5y � 3

�4x � 3y � 8

a1

a2

c1

c2c1

c2

b1

b2a1

a2

b1

b2�a1

a2

b1

b2�

DyDx

y,x

Dy,Dxyx

D.

yx

y �a1c2 � a2c1

a1b2 � a2b1

�a1

a2

c1

c2a1

a2

b1

b2x �

c1b2 � c2b1

a1b2 � a2b1

�c1

c2

b1

b2a1

a2

b1

b2

a1b2 � a2b1 � 0.

y �a1c2 � a2c1

a1b2 � a2b1

x �c1b2 � c2b1

a1b2 � a2b1

�a1x � b1y � c1

a2x � b2y � c2


8.5 APPLICATIONS OF MATRICES AND DETERMINANTS


• Use Cramer’s Rule to solve systems of linear equations.

• Use determinants to find the areas of triangles.

• Use a determinant to test forcollinear points and find an equation of a line passing through two points.

• Use matrices to encode and decode messages.


You can use Cramer’s Rule to solvereal-life problems. For instance, inExercise 69 on page 627, Cramer’sRule is used to find a quadratic modelfor the per capita consumption of bottled water in the United States.

MAFORD/istockphoto.com

Section 8.5 Applications of Matrices and Determinants 617

Cramer’s Rule generalizes easily to systems of equations in variables. The

value of each variable is given as the quotient of two determinants. The denominator is

the determinant of the coefficient matrix, and the numerator is the determinant of the

matrix formed by replacing the column corresponding to the variable (being solved for)

with the column representing the constants. For instance, the solution for in the

following system is shown.

Using Cramer’s Rule for a 2 2 System

Use Cramer’s Rule to solve the system of linear equations.

Solution

To begin, find the determinant of the coefficient matrix.

Because this determinant is not zero, you can apply Cramer’s Rule.

So, the solution is and Check this in the original system.

Now try Exercise 7.

y � �1.x � 2

y �Dy

D�

43 10

11�14

�44 � 30

�14�

14

�14� �1

x �Dx

D�

10

11

�2

�5�14

��50 � ��22�

�14��28

�14� 2

D � 43 �2

�5 � �20 � ��6� � �14

�4x � 2y � 10

3x � 5y � 11

Example 1

x3 �A3A �

a11

a21

a31

a12

a22

a32

b1

b2

b3a11

a21

a31

a12

a22

a32

a13

a23

a33�a11x1 � a12x2 � a13x3 � b1

a21x1 � a22x2 � a23x3 � b2

a31x1 � a32x2 � a33x3 � b3

x3

nn

Cramer’s Rule

If a system of linear equations in variables has a coefficient matrix with a

nonzero determinant the solution of the system is

where the th column of is the column of constants in the system of equations.

If the determinant of the coefficient matrix is zero, the system has either no

solution or infinitely many solutions.

Aii

xn �AnA. . . ,x2 �

A2A ,x1 �

A1A ,

A,Ann

Using Cramer’s Rule for a 3 3 System

Use Cramer’s Rule to solve the system of linear equations.

Solution

To find the determinant of the coefficient matrix

expand along the second row, as follows.

Because this determinant is not zero, you can apply Cramer’s Rule.

The solution is Check this in the original system as follows.

Check


Remember that Cramer’s Rule does not apply when the determinant of the coefficient

matrix is zero. This would create division by zero, which is undefined.

��45�

�45

2�45�85

3�45�

125

�

�

�

�

2��32�3

4��32�6

�

�

�

�

�

�

3��85�

245

��85�85

4��85�

325

�?

�

�?

�

�?

�

1

1

0

0

2

2

� 45, �

32, �

85�.

z �Dz

D�

�1

2

3

2

0

�4

1

0

210

��16

10� �

8

5

y �Dy

D�

�1

2

3

1

0

2

�3

1

410

��15

10� �

3

2

x �Dx

D�

102 2

0

�4

�3

1

410

�8

10�

4

5

� 10

� �2��4� � 0 � 1��2�

D � 2��1�3 2

�4

�3

4 � 0��1�4�1

3

�3

4 � 1��1�5�1

3

2

�4

��1

2

3

2

0

�4

�3

1

4�

��x

2x

3x

� 2y

� 4y

� 3z � 1

� z � 0

� 4z � 2

Example 2


Substitute into Equation 1.

Equation 1 checks. �






Area of a Triangle

Another application of matrices and determinants is finding the area of a triangle whose

vertices are given as points in a coordinate plane.

Finding the Area of a Triangle

Find the area of a triangle whose vertices are and as shown in

Figure 8.1.

Solution

Let and Then, to find the area of the

triangle, evaluate the determinant.

Using this value, you can conclude that the area of the triangle is

Choose so that the area is positive.


�3

2 square units.

� �1

2��3�

�� Area � �1

2 124 0

2

3

1

1

1� �3.

� 1��1� � 0 � 1��2�

� 1��1�223 1

1 � 0��1�324 1

1 � 1��1�424 2

3x1

x2

x3

y1

y2

y3

1

1

1 � 124 0

2

3

1

1

1�x3, y3� � �4, 3�.�x1, y1� � �1, 0�, �x2, y2� � �2, 2�,

�4, 3�,�1, 0�, �2, 2�,

Example 3

Area of a Triangle

The area of a triangle with vertices and is

where the symbol indicates that the appropriate sign should be chosen to yield

a positive area.

±

Area � ±1

2 x1

x2

x3

y1

y2

y3

1

1

1�x3, y3��x2, y2�,�x1, y1�,

x

1 2 3 4

1

2

3

(1, 0)

(2, 2)

(4, 3)

y

FIGURE 8.1


Lines in a Plane

What if the three points in Example 3 had been on the same line? What would have

happened had the area formula been applied to three such points? The answer is that the

determinant would have been zero. Consider, for instance, the three collinear points

and as shown in Figure 8.2. The area of the “triangle” that has these

three points as vertices is

The result is generalized as follows.

Testing for Collinear Points

Determine whether the points and are collinear. (See Figure 8.3.)

Solution

Letting and you have

Because the value of this determinant is not zero, you can conclude that the three points

do not lie on the same line. Moreover, the area of the triangle with vertices at these

points is square units.


��12��6� � 3

� �6.

� �2��4� � 2��6� � 1��2�

� �2��1�215 1

1� ��2��1�317 1

1� 1��1�417 1

5x1

x2

x3

y1

y2

y3

1

1

1 � �2

1

7

�2

1

5

1

1

1�x3, y3� � �7, 5�,�x1, y1� � ��2, �2�, �x2, y2� � �1, 1�,

�7, 5��2, �2�, �1, 1�,

Example 4

� 0.

�1

2�0 � 1��2� � 1��2��

1

2024 1

2

3

1

1

1 � 1

2�0��1�223 1

1� 1��1�324 1

1� 1��1�424 2

3��4, 3�,�0, 1�, �2, 2�,

Test for Collinear Points

Three points and are collinear (lie on the same line) if

and only if

x1

x2

x3

y1

y2

y3

1

1

1 � 0.

�x3, y3��x1, y1�, �x2, y2�,

x

(7, 5)

(1, 1)

(−2, −2)

−1 2

1

2

3

4

5

6

7

y

3 4 5 6 71

FIGURE 8.3

x

1 2 3 4

1

2

3

(0, 1)

(2, 2)

(4, 3)

y

FIGURE 8.2


The test for collinear points can be adapted to another use. That is, if you are given

two points on a rectangular coordinate system, you can find an equation of the line

passing through the two points, as follows.

Finding an Equation of a Line

Find an equation of the line passing through the two points and as shown

in Figure 8.4.

Solution

Let and Applying the determinant formula for the

equation of a line produces

To evaluate this determinant, you can expand by cofactors along the first row to obtain

the following.

So, an equation of the line is


Note that this method of finding the equation of a line works for all lines, including

horizontal and vertical lines. For instance, the equation of the vertical line through

and is

x � 2.

4 � 2x � 0

x22 y

0

2

1

1

1 � 0

�2, 2��2, 0�

x � 3y � 10 � 0.

x � 3y � 10 � 0

x�1��1� � y��1��3� � �1��1��10� � 0

x��1�243 1

1� y��1�3 2

�1

1

1� 1��1�4 2

�1

4

3 � 0

x

2

�1

y

4

3

1

1

1 � 0.

�x2, y2� � ��1, 3�.�x1, y1� � �2, 4�

��1, 3�,�2, 4�

Example 5

Two-Point Form of the Equation of a Line

An equation of the line passing through the distinct points and is

given by

xx1

x2

y

y1

y2

1

1

1 � 0.

�x2, y2��x1, y1�

x

(2, 4)

(−1, 3)

−1 1 2 3 4

1

2

4

5

y

FIGURE 8.4

Cryptography

A cryptogram is a message written according to a secret code. (The Greek word kryptos

means “hidden.”) Matrix multiplication can be used to encode and decode messages. To

begin, you need to assign a number to each letter in the alphabet (with 0 assigned to a

blank space), as follows.

Then the message is converted to numbers and partitioned into uncoded row matrices,

each having entries, as demonstrated in Example 6.

Forming Uncoded Row Matrices

Write the uncoded row matrices of order for the message

MEET ME MONDAY.

Solution

Partitioning the message (including blank spaces, but ignoring punctuation) into groups

of three produces the following uncoded row matrices.

Note that a blank space is used to fill out the last uncoded row matrix.

Now try Exercise 55(a).

To encode a message, use the techniques demonstrated in Section 8.3 to choose an

invertible matrix such as

and multiply the uncoded row matrices by (on the right) to obtain coded row matrices.

Here is an example.

Uncoded Matrix Encoding Matrix A Coded Matrix

�13 �26 21��1

�1

1

�2

1

�1

2

3

�4��13 5 5�

A

A � �1

�1

1

�2

1

�1

2

3

�4�

n n

M E E T M E M O N D A Y

�13 5 5� �20 0 13� �5 0 13� �15 14 4� �1 25 0�

1 3

Example 6

n

26 � Z17 � Q8 � H

25 � Y16 � P7 � G

24 � X15 � O6 � F

23 � W14 � N5 � E

22 � V13 � M4 � D

21 � U12 � L3 � C

20 � T11 � K2 � B

19 � S10 � J1 � A

18 � R19 � I0 � _



Encoding a Message

Use the following invertible matrix to encode the message MEET ME MONDAY.

Solution

The coded row matrices are obtained by multiplying each of the uncoded row matrices

found in Example 6 by the matrix as follows.

Uncoded Matrix Encoding Matrix A Coded Matrix

So, the sequence of coded row matrices is

Finally, removing the matrix notation produces the following cryptogram.

Now try Exercise 55(b).

For those who do not know the encoding matrix decoding the cryptogram found

in Example 7 is difficult. But for an authorized receiver who knows the encoding matrix

decoding is simple. The receiver just needs to multiply the coded row matrices by

(on the right) to retrieve the uncoded row matrices. Here is an example.

A�1

�13 �26 21��1

�1

0

�10

�6

�1

�8

�5

�1� � �13 5 5�

A�1

A,

A,

13 �26 21 33 �53 �12 18 �23 �42 5 �20 56 �24 23 77

��24 23 77�.�5 �20 56��18 �23 �42��33 �53 �12��13 �26 21�

�1 25 0� �1

�1

1

�2

1

�1

2

3

�4� � ��24 23 77�

�15 14 4� �1

�1

1

�2

1

�1

2

3

�4� � �5 �20 56�

�5 0 13� �1

�1

1

�2

1

�1

2

3

�4� � �18 �23 �42�

�20 0 13� �1

�1

1

�2

1

�1

2

3

�4� � �33 �53 �12�

�13 5 5� �1

�1

1

�2

1

�1

2

3

�4� � �13 �26 21�

A,

A � �1

�1

1

�2

1

�1

2

3

�4�

Example 7

Coded Uncoded


Decoding a Message

Use the inverse of the matrix

to decode the cryptogram

Solution

First find by using the techniques demonstrated in Section 8.3. is the decoding

matrix. Then partition the message into groups of three to form the coded row matrices.

Finally, multiply each coded row matrix by (on the right).

Coded Matrix Decoding Matrix Decoded Matrix

So, the message is as follows.


M E E T M E M O N D A Y

�13 5 5� �20 0 13� �5 0 13� �15 14 4� �1 25 0�

��24 23 77� ��1

�1

0

�10

�6

�1

�8

�5

�1� � �1 25 0�

�5 �20 56� ��1

�1

0

�10

�6

�1

�8

�5

�1� � �15 14 4�

�18 �23 �42� ��1

�1

0

�10

�6

�1

�8

�5

�1� � �5 0 13�

�33 �53 �12� ��1

�1

0

�10

�6

�1

�8

�5

�1� � �20 0 13�

�13 �26 21� ��1

�1

0

�10

�6

�1

�8

�5

�1� � �13 5 5�

A�1

A�1

A�1A�1

13 �26 21 33 �53 �12 18 �23 �42 5 �20 56 �24 23 77.

A � �1

�1

1

�2

1

�1

2

3

�4�

Example 8

Cryptography Use your school’s library, the Internet, or some other referencesource to research information about another type of cryptography. Write a shortparagraph describing how mathematics is used to code and decode messages.

CLASSROOM DISCUSSION

HISTORICAL NOTE

During World War II, Navajo soldiers created a code usingtheir native language to sendmessages between battalions.Native words were assigned

to represent characters in theEnglish alphabet, and they

created a number of expressionsfor important military terms,

such as iron-fish to mean submarine. Without the NavajoCode Talkers, the Second World

War might have had a very different outcome.

Bettmann/Corbis



VOCABULARY: Fill in the blanks.

1. The method of using determinants to solve a system of linear equations is called ________ ________.

2. Three points are ________ if the points lie on the same line.

3. The area of a triangle with vertices and is given by ________.

4. A message written according to a secret code is called a ________.

5. To encode a message, choose an invertible matrix and multiply the ________ row matrices by

(on the right) to obtain ________ row matrices.

6. If a message is encoded using an invertible matrix then the message can be decoded by multiplying

the coded row matrices by ________ (on the right).


A,

AA

�x3, y3��x2, y2�,�x1, y1�,A

In Exercises 7–16, use Cramer’s Rule to solve (if possible) thesystem of equations.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

In Exercises 17–20, use a graphing utility and Cramer’s Ruleto solve (if possible) the system of equations.

17. 18.

19. 20.

In Exercises 21–32, use a determinant and the given verticesof a triangle to find the area of the triangle.

21. 22.

23. 24.

25. 26.

27. 28.

29. 30.

31. 32.

In Exercises 33 and 34, find a value of such that the trianglewith the given vertices has an area of 4 square units.

33. 34.

In Exercises 35 and 36, find a value of such that the trianglewith the given vertices has an area of 6 square units.

35.

36.

37. AREA OF A REGION A large region of forest has

been infested with gypsy moths. The region is roughly

triangular, as shown in the figure on the next page. From

the northernmost vertex of the region, the distances to

the other vertices are 25 miles south and 10 miles east

(for vertex ), and 20 miles south and 28 miles east (for

vertex ). Use a graphing utility to approximate the

number of square miles in this region.

C

B

A

�1, 0�, �5, �3�, ��3, y��2, �3�, �1,�1�, ��8, y�

y

��4, 2�, ��3, 5�, ��1, y��5, 1�, �0, 2�, ��2, y�

y

�92, 0�, �2, 6�, �0, �

32��4, 2�, �0,

72�, �3, �

12�

��2, 4�, �1, 5�, �3, �2��3, 5�, �2, 6�, �3, �5��0, �2�, ��1, 4�, �3, 5��2, 4�, �2, 3�, ��1, 5�

(6, −1)

(6, 10)

(−4, −5)

−8

4

8

y

x

(4, 3)

( )

1

1

2

3

4

1

20,

( )5

2, 0

x

y

2 3 4

(1, 6)

(−2, 1)

4

2

6

y

x

(3, −1)

−2

(0, 4)

(2, −3)(−2, −3)

−4 2−2

4

y

x

4

(4, 5)

(5, −2)

(0, 0)

−1 4

−2

1

2

3

4

5

y

x

1 6x

(0, 0)

(1, 5)

(3, 1)

5

1

2

3

4

5

y

4321

�3x

2x

x

�

�

�

y

y

y

�

�

�

3z

2z

z

�

�

�

1

�4

5�2x

x

3x

�

�

�

y

2y

y

�

�

�

z

z

z

�

�

�

5

1

4

�x � 2y � z �

2x � 2y � 2z �

�x � 3y � 4z �

�7

�8

8�

3x � 3y � 5z � 1

3x � 5y � 9z � 2

5x � 9y � 17z � 4

�5x� 4y� z�

�x� 2y� 2z�

3x� y� z�

�14

10

1�

x� 2y� 3z�

�2x� y� z�

3x� 3y� 2z�

�3

6

�11

�4x � 2y � 3z �

2x � 2y � 5z �

8x � 5y � 2z �

�2

16

4�

4x � y � z �

2x � 2y � 3z �

5x � 2y � 6z �

�5

10

1

� 2.4x � 1.3y �

�4.6x � 0.5y �

14.63

�11.51��0.4x � 0.8y � 1.6

0.2x � 0.3y � 2.2

� 6x � 5y �

�13x � 3y �

17

�76�3x � 2y �

6x � 4y �

�2

4

�4x6x

�

�

3y

9y

�

�

�10

12��7x

3x

�

�

11y

9y

�

�

�1

9


FIGURE FOR 37

38. AREA OF A REGION You own a triangular tract of

land, as shown in the figure. To estimate the number

of square feet in the tract, you start at one vertex, walk

65 feet east and 50 feet north to the second vertex, and

then walk 85 feet west and 30 feet north to the third

vertex. Use a graphing utility to determine how many

square feet there are in the tract of land.

In Exercises 39–44, use a determinant to determine whetherthe points are collinear.

39. 40.

41. 42.

43. 44.

In Exercises 45 and 46, find such that the points arecollinear.

45. 46.

In Exercises 47–52, use a determinant to find an equation ofthe line passing through the points.

47. 48.

49. 50.

51. 52.

In Exercises 53 and 54, (a) write the uncoded rowmatrices for the message. (b) Then encode the message usingthe encoding matrix.

Message Encoding Matrix

53. COME HOME SOON

54. HELP IS ON THE WAY

In Exercises 55 and 56, (a) write the uncoded row matrices for the message. (b) Then encode the message usingthe encoding matrix.


55. CALL ME TOMORROW

56. PLEASE SEND MONEY

In Exercises 57–60, write a cryptogram for the message usingthe matrix

57. LANDING SUCCESSFUL

58. ICEBERG DEAD AHEAD

59. HAPPY BIRTHDAY

60. OPERATION OVERLOAD

In Exercises 61–64, use to decode the cryptogram.

61.

11 21 64 112 25 50 29 53 23 46

40 75 55 92

62.

85 120 6 8 10 15 84 117 42 56 90

125 60 80 30 45 19 26

63.

9 38 28

25 41 21 31 9 �4�5�64�80

�19�9�19�19�9�1

A � �1

1

�6

�1

0

2

0

�1

3�

A � �2

3

3

4�

A � �1

3

2

5�A 1

A ! [1

3

1

2

7

4

2

9

7]

A.

�4

�3

3

2

�3

2

1

�1

1�

�1

1

�6

�1

0

2

0

�1

3�

1 " 3

��2

�1

3

1��1

3

2

5�

1 " 2

� 23, 4�, �6, 12��1

2, 3�, � 52, 1�

�10, 7�, ��2, �7��4, 3�, �2, 1��0, 0�, ��2, 2��0, 0�, �5, 3�

��6, 2�, ��5, y�, ��3, 5��2, �5�, �4, y�, �5, �2�

y

�2, 3�, �3, 3.5�, ��1, 2��0, 2�, �1, 2.4�, ��1, 1.6��0,

12�, �2, �1�, ��4,

72��2, �

12�, ��4, 4�, �6, �3�

�3, �5�, �6, 1�, �4, 2��3, �1�, �0, �3�, �12, 5�

65

85

50

30

N

S

EW

A

C

B

25

28

10

20

N

S

EW

64.

112 83 19 13 72 61 95

71 20 21 38 35 36 42 32

In Exercises 65 and 66, decode the cryptogram by using theinverse of the matrix

65. 20 17 1

62 143 181

66. 13 61 112 106 11

24 29 65 144 172

67. The following cryptogram was encoded with a

matrix.

8 21 5 10 5 25 5 19

6 20 40 1 16

The last word of the message is _RON. What is the

message?

68. The following cryptogram was encoded with a

matrix.

5 2 25 11 32 14 8

38 19 37 16

The last word of the message is _SUE. What is the

message?

69. DATA ANALYSIS: BOTTLED WATER The table

shows the per capita consumption of bottled water

(in gallons) in the United States from 2000 through

2007. (Source: Economic Research Service, U.S.

Department of Agriculture)

(a) Use the technique demonstrated in Exercises 77–80

in Section 7.3 to create a system of linear equations

for the data. Let represent the year, with

corresponding to 2000.

(b) Use Cramer’s Rule to solve the system from part (a)

and find the least squares regression parabola

(c) Use a graphing utility to graph the parabola from

part (b).

(d) Use the graph from part (c) to estimate when the

per capita consumption of bottled water will exceed

35 gallons.

70. HAIR PRODUCTS A hair product company sells

three types of hair products for $30, $20, and $10 per

unit. In one year, the total revenue for the three products

was $800,000, which corresponded to the sale of 40,000

units. The company sold half as many units of the $30

product as units of the $20 product. Use Cramer’s Rule

to solve a system of linear equations to find how many

units of each product were sold.

EXPLORATION

TRUE OR FALSE? In Exercises 71–74, determine whetherthe statement is true or false. Justify your answer.

71. In Cramer’s Rule, the numerator is the determinant of

the coefficient matrix.

72. You cannot use Cramer’s Rule when solving a system of

linear equations if the determinant of the coefficient

matrix is zero.

73. In a system of linear equations, if the determinant of the

coefficient matrix is zero, the system has no solution.

74. The points and are collinear.

75. WRITING Use your school’s library, the Internet, or

some other reference source to research a few current

real-life uses of cryptography. Write a short summary of

these uses. Include a description of how messages are

encoded and decoded in each case.

77. Use determinants to find the area of a triangle with

vertices and Confirm your

answer by plotting the points in a coordinate plane and

using the formula

Area �12�base��height�.

�7, 5�.�7, �1�,�3, �1�,

�3, 11��0, 2�,��5, �13�,

y � at2 � bt � c.

t � 0t

y

�19�19�13

��15�15�7�2

2 2

�18�18�1

�13�13�10�15

2 2

�131�73�17�59�9

�65�25�104�56�12�15

A ! [1

3

1

2

7

4

2

9

7]

A.

�48�23�118

�76�25�140

A � �3

0

4

�4

2

�5

2

1

3�


Year Consumption, y

2000

2001

2002

2003

2004

2005

2006

2007

16.7

18.2

20.1

21.6

23.2

25.5

27.7

29.1

76. CAPSTONE

(a) State Cramer’s Rule for solving a system of linear

equations.

(b) At this point in the text, you have learned several

methods for solving systems of linear equations.

Briefly describe which method(s) you find easiest

to use and which method(s) you find most

difficult to use.


CHAPTER SUMMARY8

What Did You Learn? Explanation/Examples ReviewExercises

Sec

tio

n 8

.2Sec

tio

n 8

.3Sec

tio

n 8

.1

Write matrices and identify their

orders (p. 570).

Perform elementary row

operations on matrices (p. 572).

Use matrices and Gaussian

elimination to solve systems of

linear equations (p. 575).

Use matrices and Gauss-Jordan

elimination to solve systems of

linear equations (p. 577).

Decide whether two matrices are

equal (p. 584).

Add and subtract matrices and

multiply matrices by scalars

(p. 585).

Multiply two matrices (p. 589).

Use matrix operations to model and

solve real-life problems (p. 592).

Verify that two matrices are

inverses of each other (p. 599).


1. Interchange two rows.

2. Multiply a row by a nonzero constant.

3. Add a multiple of a row to another row.


1. Write the augmented matrix of the system of linear

equations.

2. Use elementary row operations to rewrite the augmented

matrix in row-echelon form.

3. Write the system of linear equations corresponding to

the matrix in row-echelon form, and use back-substitution

to find the solution.

Gauss-Jordan elimination continues the reduction process

on a matrix in row-echelon form until a reduced row-echelon

form is obtained. (See Example 8.)

Two matrices are equal if their corresponding entries are

equal.

Definition of Matrix Addition

If and are matrices of order their

sum is the matrix given by

Definition of Scalar Multiplication

If is an matrix and is a scalar, the scalar

multiple of by is the matrix given by

Matrix Multiplication

If is an matrix and is an

matrix, the product is an matrix

where

Matrix operations can be used to find the total cost of

equipment for two softball teams. (See Example 12.)

Inverse of a Square Matrix

Let be an matrix and let be the identity

matrix. If there exists a matrix such that

then is the inverse of A.A�1

AA�1 � In � A�1A

A�1

n nInn nA

cij � ai1b1j � ai2b2j � ai3b3j �. . . � ainbnj.

AB � �cij�m pAB

n pB � �bij�m nA � �aij�

cA � �cij�m ncA

cm nA � �aij�

A � B � �aij � bij�.m n

m n,B � �bij�A � �aij�

2 13 21 32 2

��1

4

1

7� ��2 3 0� �4

5

�2

�3

0

1� � 8

�8�1–8

9, 10

11–28

29–36

37–40

41–54

55–68

69–72

73–76

Chapter Summary 629

What Did You Learn? Explanation/Examples ReviewExercises

Sec

tio

n 8

.5Sec

tio

n 8

.4Sec

tio

n 8

.3

Use Gauss-Jordan elimination

to find the inverses of matrices

(p. 600).

Use a formula to find the inverses

of matrices (p. 603).

Use inverse matrices to solve

systems of linear equations

(p. 604).

Find the determinants of

matrices (p. 608).

Find minors and cofactors of

square matrices (p. 610).

Find the determinants of square

matrices (p. 611).

Use Cramer’s Rule to solve systems

of linear equations (p. 616).

Use determinants to find the areas

of triangles (p. 619).

Use a determinant to test for

collinear points and find an

equation of a line passing

through two points (p. 620).

Use matrices to encode and

decode messages (p. 622).

2 2

2 2

Finding an Inverse Matrix

Let be a square matrix of order

1. Write the matrix that consists of the given matrix

on the left and the identity matrix on the right to

obtain

2. If possible, row reduce to using elementary row

operations on the entire matrix The result will be the

matrix If this is not possible, is not invertible.

3. Check your work to see that

If and then

If is an invertible matrix, the system of linear equations

represented by has a unique solution given by

The determinant of the matrix is given by

If is a square matrix, the minor of the entry is the

determinant of the matrix obtained by deleting the th row

and th column of The cofactor of the entry is

If is a square matrix (of order or greater), the

determinant of is the sum of the entries in any row (or

column) of multiplied by their respective cofactors.

Cramer’s Rule uses determinants to write the solution of a

system of linear equations.

The area of a triangle with vertices and

is

where the symbol indicates that the appropriate sign

should be chosen to yield a positive area.

Three points and are collinear (lie

on the same line) if and only if

The inverse of a matrix can be used to decode a cryptogram.

(See Example 8.)

x1

x2

x3

y1

y2

y3

1

1

1 � 0.

�x3, y3��x2, y2�,�x1, y1�,

±

Area � ±1

2x1

x2

x3

y1

y2

y3

1

1

1�x3, y3�

�x2, y2�,�x1, y1�,

A

A

2 2A

Cij � ��1�i� jMij.

aijCijA.j

i

aijMijA

det�A� � A � a1

a2

b1

b2 � a1b2 � a2b1.

A � �a1

a2

b1

b2�

X � A�1B.

AX � B

A

A�1 �1

ad � bc�d

�c

�b

a�.ad � bc � 0,A � �ac

b

d�

AA�1 � I � A�1A.

A�I A�1�.�A I�.IA

�A I�.In n

An 2n

n.A

77–84

85–92

93–110

111–114

115–118

119–128

129–132

133–136

137–142

143–146

In Exercises 1–4, determine the order of the matrix.

1. 2.

3. 4.

In Exercises 5 and 6, write the augmented matrix for thesystem of linear equations.

5. 6.

In Exercises 7 and 8, write the system of linear equations represented by the augmented matrix. (Use variables and if applicable.)

7.

8.

In Exercises 9 and 10, write the matrix in row-echelon form.(Remember that the row-echelon form of a matrix is notunique.)

9. 10.

In Exercises 11–14, write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve the system. (Use variables and )

11.

12.

13.

14.

In Exercises 15–28, use matrices and Gaussian elimination withback-substitution to solve the system of equations (if possible).

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

27.

28.

In Exercises 29–34, use matrices and Gauss-Jordan eliminationto solve the system of equations.

29. 30.

31.

32.

33.

34.

��3x � y � 7z �

5x � 2y � z �

�x � y � 4z �

�20

34

�8

�2x � y � 9z �

�x � 3y � 4z �

5x � 2y � z �

�8

�15

17

�4x � 4y � 4z � 5

4x � 2y � 8z � 1

5x � 3y � 8z � 6

��x � y � 2z �

2x � 3y � z �

5x � 4y � 2z �

1

�2

4

� x

3x

�x

�

�

�

3y

y

y

�

�

�

z

z

3z

�

�

�

2

�6

�2� xx

2x

�

�

�

2y

y

y

�

�

�

z

z

3z

�

�

�

3

�3

10

�x

4x

2x

�

�

2y �

�3y �

4y �

�

3z

z

z

w � 3

� 0

� 2w � 0

� 3

�2x

3x

x

�

�

y

�2y

3y

�

�

�

�

z

3z

2z

z

�

� w �

� 2w �

� 3w �

6

9

�11

14

�2x � 3y � 3z � 3

6x � 6y � 12z � 13

12x � 9y � z � 2�

2x � 3y � z �

2x � 3y � 3z �

4x � 2y � 3z �

10

22

�2

�x � 2y � 6z �

2x � 5y � 15z �

3x � y � 3z �

1

4

�6�

2x � y

2x � 2y

2x � y

� 2z � 4

� 5

� 6z � 2

� x

2x

�x

�

�

�

2y

y

3y

�

�

�

z

2z

2z

�

�

�

4

�24

20� x

2x

�x

�

�

�

2y

y

3y

�

�

�

z

2z

2z

�

�

�

7

�4

�3

��x2x

�

�

2y

4y

�

�

3

�6��x2x

�

�

2y

4y

�

�

3

6

�0.2x � 0.1y �

0.4x � 0.5y �

0.07

�0.01�0.3x � 0.1y � �0.13

0.2x � 0.3y � �0.25

�2x � 5y � 2

3x � 7y � 1� 5x � 4y �

�x � y �

2

�22

�1

0

0

�8

1

0

0

�1

1

�2

�7

1�

�1

0

0

�5

1

0

4

2

1

1

3

4�

�1

0

0

3

1

0

�9

�1

1

4

10

�2�

�1

0

0

2

1

0

3

�2

1

9

2

0�

z.y,x,

�4

3

�2

8

�1

10

16

2

12��

0

1

2

1

2

2

1

3

2�

�13

1

4

16

21

10

7

8

�4

3

5

3

2

12

�1�

�5

4

9

1

2

4

7

0

2

�9

10

3�

w,z,y,x,

�8x � 7y � 4z � 12

3x � 5y � 2z � 20

5x � 3y � 3z � 26

�3x � 10y � 15

5x � 4y � 22

�6 2 �5 8 0��3�

� 3

�2

�1

7

0

1

6

4��4

0

5�

8.1


REVIEW EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises.8

Review Exercises 631

In Exercises 35 and 36, use the matrix capabilities of a graphingutility to reduce the augmented matrix corresponding to thesystem of equations, and solve the system.

35.

36.

In Exercises 37– 40, find and

37.

38.

39.

40.

In Exercises 41– 44, if possible, find (a) (b) (c) and (d)

41.

42.

43.

44.

In Exercises 45–48, perform the matrix operations. If it is notpossible, explain why.

45.

46.

47.

48.

In Exercises 49 and 50, use the matrix capabilities of a graphingutility to evaluate the expression.

49.

50.

In Exercises 51–54, solve for in the equation, given

and

51. 52.

53. 54.

In Exercises 55–58, find if possible.

55.

56.

57.

58.

In Exercises 59– 66, perform the matrix operations, if possible.If it is not possible, explain why.

59.

60.

61.

62. �1

0

0

3

2

0

2

�4

3��

4

0

0

�3

3

0

2

�1

2�

�1

2

5

�4

6

0��6

�2

8

4

0

0�

�1

2

5

�4

6

0� �6

4

�2

0

8

0�

�1

5

6

2

�4

0��6

4

�2

0

8

0�

B � ��1

4

8�A � �6 �5 7�,

B � � 4

20

12

40�A � �5

�7

11

4

2

2�,

B � �4

20

15

12

40

30�A � �

5

�7

11

4

2

2�,B � ��3

12

10

8�A � �2

3

�2

5�,AB,

2A � 5B � 3X3X � 2A � B

6X � 4A � 3BX � 2A � 3B

B ! �1

2

4

2

1

4�.A ! �

4

1

3

0

5

2�

X

�5�2

7

8

0

�2

2� � 4�

4

6

�1

�2

11

3�

3 �8

1

�2

3

5

�1� � 6 �4

2

�2

7

�3

6�

��8

�2

0

�1

4

�6

8

12

0� � 5�

�2

3

6

0

�1

12

�4

1

�8�

�2�1

5

6

2

�4

0� � 8�

7

1

1

1

2

4�

��11

�7

16

�2

19

1� � �6

8

�2

0

�4

10�

� 7

�1

3

5� � �10

14

�20

�3�

B � ��1

4

8�A � �6 �5 7�,

B � �0

4

20

3

12

40�A � �

5

�7

11

4

2

2�,

B � �3

15

20

11

25

29�A � �

4

�6

10

3

1

1�,B � ��3

12

10

8�A � �2

3

�2

5�,A # 3B.4A,

A B,A # B,

��9

012x

4

�3

�1

x � 10

7

1

�5

2y

0��

�9

0

6

4

�3

�1

2

7

1

�5

�4

0�

�x � 3

0

�2

�4

�3

y � 5

4y

2

6x� � �

5x � 1

0

�2

�4

�3

16

44

2

6�

��1

x

�4

0

5

y� � �

�1

8

�4

0

5

0�

��1

y

x

9� � ��1

�7

12

9�y.x8.2

�4x � 12y � 2z �

x � 6y � 4z �

x � 6y � z �

�2x � 10y � 2z �

20

12

8

�10

�3x � y � 5z � 2w �

x � 6y � 4z � w �

5x � y � z � 3w �

4y � z � 8w �

�44

1

�15

58


63.

64.

65.

66.

In Exercises 67 and 68, use the matrix capabilities of a graphingutility to find the product.

67.

68.

69. MANUFACTURING A tire corporation has three

factories, each of which manufactures two models of

tires. The number of units of model produced at factory

in one day is represented by in the matrix

Find the production levels if production is decreased by 5%.

70. MANUFACTURING A power tool company has four

manufacturing plants, each of which produces three

types of cordless power tools. The number of units of

cordless power tool produced at plant in one day is

represented by in the matrix

Find the production levels if production is increased by 20%.

71. MANUFACTURING An electronics manufacturing

company produces three different models of headphones

that are shipped to two warehouses. The number of

units of model that are shipped to warehouse is

represented by in the matrix

The price per unit is represented by the matrix


72. CELL PHONE CHARGES The pay-as-you-go charges

(in dollars per minute) of two cellular telephone companies

for calls inside the coverage area, regional roaming calls,

and calls outside the coverage area are represented by

Company

A B

Coverage area

Each month, you plan to use 120 minutes on calls inside

the coverage area, 80 minutes on regional roaming calls,

and 20 minutes on calls outside the coverage area.

(a) Write a matrix that represents the times spent on

the phone for each type of call.

(b) Compute and interpret the result.

In Exercises 73–76, show that is the inverse of

73.

74.

75.

76.

In Exercises 77–80, find the inverse of the matrix (if it exists).

77. 78.

79. 80.

In Exercises 81–84, use the matrix capabilities of a graphingutility to find the inverse of the matrix (if it exists).

81. 82.

83. 84. �8

4

1

�1

0

�2

2

4

2

0

1

1

8

�2

4

1��

1

4

3

�1

3

4

4

2

1

2

1

�1

6

6

2

�2�

�1

2

�1

4

�3

18

6

1

16��

�1

3

1

�2

7

4

�2

9

7�

�0

�5

7

�2

�2

3

1

�3

4��

2

�1

2

0

1

�2

3

1

1�

��3

2

�5

3��6

�5

5

4�

B � ��2

�3

2

1

1

�2

12

12

�12

�A � �1

�1

8

�1

0

�4

0

�1

2�,B � �

�2

3

2

�3

3

4

1

�1

�1�A � �

1

1

6

1

0

2

0

1

3�,B � � �2

�11

1

5�A � � 5

11

�1

�2�,B � ��2

7

�1

4�A � ��4

7

�1

2�,A.B8.3

TC

T

Inside

Regional roaming

Outside

C � �0.07

0.10

0.28

0.095

0.08

0.25�

C.

BA

B � �$79.99 $109.95 $189.99�.

A � �8200

6500

5400

7400

9800

4800�.

aij

ji

A � �80

50

90

70

30

60

90

80

100

40

20

50�.

aij

ji

A � �80

40

120

100

140

80�.

aijj

i

��2

4

3

�2

10

2��1

�5

3

1

2

2�

�4

11

12

1

�7

3��3

2

�5

�2

6

�2�

�3�1

4

�1

2��0

1

3

2� �1

5

0

�3� �2

6

1

0�� 4

�3

2

1� � ��2

0

4

4�

�4 �2 6� ��2

0

2

1

�3

0�

�1

0

1

2

4

1

�1

�2

3��1 �1 2�

Review Exercises 633

In Exercises 85–92, use the formula below to find the inverseof the matrix, if it exists.

85. 86.

87. 88.

89. 90.

91. 92.

In Exercises 93–104, use an inverse matrix to solve (ifpossible) the system of linear equations.

93. 94.

95. 96.

97. 98.

99. 100.

101.

102.

103.

104.

In Exercises 105–110, use the matrix capabilities of a graphingutility to solve (if possible) the system of linear equations.

105. 106.

107. 108.

109.

110.

In Exercises 111–114, find the determinant of the matrix.

111. 112.

113. 114.

In Exercises 115–118, find all (a) minors and (b) cofactors ofthe matrix.

115. 116.

117. 118.

In Exercises 119–128, find the determinant of the matrix.Expand by cofactors on the row or column that appears tomake the computations easiest.

119. 120.

121. 122.

123. 124.

125. 126.

127. 128.

In Exercises 129–132, use Cramer’s Rule to solve (ifpossible) the system of equations.

129. 130.

131.

��2x � 3y � 5z �

4x � y � z �

�x � 4y � 6z �

�11

�3

15

�3x � 8y �

9x � 5y �

�7

37� 5x � 2y �

�11x � 3y �

6

�23

8.5

��5

0

�3

1

6

1

4

6

0

�1

�5

0

0

2

1

3��

3

0

6

0

0

8

1

3

�4

1

8

�4

0

2

2

1�

�1

4

2

0

�2

1

3

�2

1

4

3

�4

2

1

0

2��

1

1

2

2

2

2

�4

0

�1

�4

�3

0

0

1

1

0�

�1

�4

0

1

1

1

4

2

�1��

�2

�6

5

4

0

3

1

2

4�

��1

2

�5

�2

3

�1

1

0

3��

4

2

1

1

3

�1

�1

2

0�

�0

0

�1

1

1

�1

�2

2

3��

�2

2

�1

0

�1

1

0

0

�3�

�8

6

�4

3

5

1

4

�9

2��

3

�2

1

2

5

8

�1

0

6�

�3

5

6

�4��2

7

�1

4�

�14

12

�24

�15��50

10

�30

5��9

7

11

�4��8

2

5

�4�8.4

�x � 3y � 2z �

�2x � 7y � 3z �

x � y � 3z �

8

�19

3

��3x � 3y � 4z �

y � z �

4x � 3y � 4z �

2

�1

�1

�5x

2x

�

�

10y

y

�

�

7

�98�65x

�125 x

�

�

47y

127 y

�

�

65

�175

� x � 3y �

�6x � 2y �

23

�18� x � 2y � �1

3x � 4y � �5

�3x � y � 5z �

�x � y � 6z �

�8x � 4y � z �

�14

8

44

��2x � y � 2z �

�x � 4y � z �

�y � z �

�13

�11

0

��x � 4y � 2z �

2x � 9y � 5z �

�x � 5y � 4z �

12

�25

10

�3x � 2y � z �

x � y � 2z �

5x � y � z �

6

�1

7

�3.5x

2.5x

�

�

4.5y

7.5y

�

�

8

25�0.3x

0.4x

�

�

0.7y

0.6y

�

�

10.2

7.6

��56x

4x

�

�

38y

3y

�

�

�2

0�12x

�3x

�

�

13y

2y

�

�

2

0

� 4x � 2y �

�19x � 9y �

�10

47��3x � 10y �

5x � 17y �

8

�13

� 5x � y �

�9x � 2y �

13

�24��x � 4y �

2x � 7y �

8

�5

�1.6

1.2

�3.2

�2.4�� 0.5

�0.2

0.1

�0.4��

34

�45

52

�83��

12

310

20

�6��18

�6

�15

�5��12

10

6

�5��10

7

4

3��7

�8

2

2�

A 1!

1

ad bc[ d

c

b

a]


132.

In Exercises 133–136, use a determinant and the given vertices of a triangle to find the area of the triangle.

133. 134.

135. 136.

In Exercises 137 and 138, use a determinant to determinewhether the points are collinear.

137.

138.

In Exercises 139–142, use a determinant to find an equationof the line passing through the points.

139. 140.

141. 142.

In Exercises 143 and 144, (a) write the uncoded rowmatrices for the message, and (b) encode the message usingthe encoding matrix.


143. LOOK OUT BELOW

144. HEAD DUE WEST

In Exercises 145 and 146, decode the cryptogram by using theinverse of the matrix

145. 5 11 2 370 265 225 57 48 33 32

15 20 245 171 147

146. 145 105 92 264 188 160 23 16 15

129 84 78 9 8 5 159 118 100 219

152 133 370 265 225 105 84 63

EXPLORATION


147. It is possible to find the determinant of a matrix.

148.

149. Use the matrix capabilities of a graphing utility to find

the inverse of the matrix

What message appears on the screen? Why does the

graphing utility display this message?

150. Under what conditions does a matrix have an inverse?

151. WRITING What is meant by the cofactor of an entry

of a matrix? How are cofactors used to find the

determinant of the matrix?

152. Three people were asked to solve a system of

equations using an augmented matrix. Each person

reduced the matrix to row-echelon form. The reduced

matrices were

and

Can all three be right? Explain.

153. THINK ABOUT IT Describe the row-echelon form of

an augmented matrix that corresponds to a system of

linear equations that has a unique solution.

154. Solve the equation for

2 � !3

5

�8 � ! � 0

!.

�1

0

2

0

3

0�.

�1

0

0

1

1

1�,�1

0

2

1

3

1�,

A � � 1

�2

�3

6�.

a11

a21

a31

a12

a22

a32

a13

a23

a33 � a11

a21

c1

a12

a22

c2

a13

a23

c3 a11

a21

a31 � c1

a12

a22

a32 � c2

a13

a23

a33 � c3 �4 5

��

��

��

��

��

A ! [ 5

10

8

4

7

6

3

6

5].

�1

3

�1

2

7

�4

2

9

�7�

�2

3

�6

�2

0

2

0

�3

3�

1 " 3

�0.7, 3.2��0.8, 0.2�,��52, 3�, � 7

2, 1��2, 5�, �6, �1��4, 0�, �4, 4�

�0, �5�, ��2, �6�, �8, �1��1, 7�, �3, �9�, ��3, 15�

(4, 2)

4, − 1 2 3

y

1

2( (

, 1 3

2( (1

2

3

x

(−2, 3)

(1, −4)

(0, 5)

2

−2

−4

42−2−4

6

y

x

(0, 6)

(4, 0)

(−4, 0)

2

42−2−4

6

y

x

(5, 8)

(5, 0)

(1, 0)

2

4

−24 6 8

6

8

y

x

�5x � 2y � z �

3x � 3y � z �

2x � y � 7z �

15

�7

�3

Chapter Test 635

CHAPTER TEST See www.CalcChat.com for worked-out solutions to odd-numbered exercises.8

Take this test as you would take a test in class. When you are finished, check your workagainst the answers given in the back of the book.

In Exercises 1 and 2, write the matrix in reduced row-echelon form.

1. 2.

3. Write the augmented matrix corresponding to the system of equations and solve the

system.

4. Find (a) (b) 3 (c) and (d) (if possible).

In Exercises 5 and 6, find the inverse of the matrix (if it exists).

5. 6.

7. Use the result of Exercise 5 to solve the system.

In Exercises 8–10, evaluate the determinant of the matrix.

8. 9. 10.

In Exercises 11 and 12, use Cramer’s Rule to solve (if possible) the system of equations.

11. 12.

13. Use a determinant to find the area of the triangle in the figure.

14. Find the uncoded row matrices for the message KNOCK ON WOOD. Then

encode the message using the matrix below.

15. One hundred liters of a 50% solution is obtained by mixing a 60% solution with a

20% solution. How many liters of each solution must be used to obtain the desired

mixture?

A � �1

1

6

�1

0

�2

0

�1

�3�

A

1 3

� 6x � y � 2z �

�2x � 3y � z �

4x � 4y � z �

�4

10

�18

� 7x � 6y �

�2x � 11y �

9

�49

�6

3

1

�7

�2

5

2

0

1��

52

�8

13465��6

10

4

12�

��4x � 3y � 6

5x � 2y � 24

��2

2

4

4

1

�2

�6

0

5��4

5

3

�2�

B � � 5

�5

0

�1�A � � 6

�5

5

�5�,

AB3A � 2B,A,A � B,

�4x � 3y � 2z �

�x � y � 2z �

3x � y � 4z �

14

�5

8

�1

�1

1

3

0

1

1

2

�1

1

�1

�3

2

�3

1

4��

1

6

5

�1

2

3

5

3

�3�

−2−4

4

6

−24

(−5, 0)

(4, 4)

(3, 2)x

y

2

FIGURE FOR 13

636

PROOFS IN MATHEMATICS

Proofs without words are pictures or diagrams that give a visual understanding of why

a theorem or statement is true. They can also provide a starting point for writing a

formal proof. The following proof shows that a determinant is the area of a

parallelogram.

The following is a color-coded version of the proof along with a brief explanation

of why this proof works.

Area of Area of orange Area of yellow Area of blue

Area of pink Area of white quadrilateral

Area of Area of orange Area of pink Area of green

quadrilateral

Area of Area of white quadrilateral Area of blue Area of yellow

Area of green quadrilateral

Area of Area of

From “Proof Without Words” by Solomon W. Golomb, Mathematics Magazine, March 1985, Vol. 58, No. 2,

pg. 107. Reprinted with permission.

!" ��

# �# ��$ �

# �# �! �

# �

# �# �# �" �

ac b

d � ad � bc � !" ! � ! &! � !$!

(0, d)

(0, 0)

(a, d)

(a, b)

(a, 0)

(a + c, d)

(a + c, b + d)(a, b + d)

ac b

d � ad � bc � !" ! � ! &! � !$!

(0, d)

(0, 0)

(a, d)

(a, b)

(a, 0)

(a + c, d)

(a + c, b + d)(a, b + d)

2 2

637

1. The columns of matrix show the coordinates of the

vertices of a triangle. Matrix is a transformation matrix.

(a) Find and Then sketch the original triangle

and the two transformed triangles. What transformation

does represent?

(b) Given the triangle determined by describe the

transformation process that produces the triangle

determined by and then the triangle determined by

2. The matrices show the number of people (in thousands)

who lived in each region of the United States in 2000 and

the number of people (in thousands) projected to live

in each region in 2015. The regional populations are

separated into three age categories. (Source: U.S.

Census Bureau)

2000

0–17 18–64 65 +

2015

0–17 18–64 65 +

(a) The total population in 2000 was approximately

281,422,000 and the projected total population in

2015 is 322,366,000. Rewrite the matrices to give the

information as percents of the total population.

(b) Write a matrix that gives the projected change in the

percent of the population in each region and age

group from 2000 to 2015.

(c) Based on the result of part (b), which region(s) and

age group(s) are projected to show relative growth

from 2000 to 2015?

3. Determine whether the matrix is idempotent. A square

matrix is idempotent if

(a) (b)

(c) (d)

4. Let

(a) Show that where is the identity

matrix of order 2.

(b) Show that

(c) Show in general that for any square matrix satisfying

the inverse of is given by

5. Two competing companies offer satellite television to a

city with 100,000 households. Gold Satellite System has

25,000 subscribers and Galaxy Satellite Network has

30,000 subscribers. (The other 45,000 households do not

subscribe.) The percent changes in satellite subscriptions

each year are shown in the matrix below.

Percent Changes

From From From Non-

Gold Galaxy subscriber

(a) Find the number of subscribers each company will

have in 1 year using matrix multiplication. Explain

how you obtained your answer.

(b) Find the number of subscribers each company will

have in 2 years using matrix multiplication. Explain


(c) Find the number of subscribers each company will

have in 3 years using matrix multiplication. Explain


(d) What is happening to the number of subscribers to

each company? What is happening to the number of

nonsubscribers?

6. Find such that the matrix is equal to its own inverse.

7. Find such that the matrix is singular.

8. Find an example of a singular matrix satisfying

A2 � A.

2 2

A � � 4

�2

x

�3�

x

A � � 3

�2

x

�3�

x

�0.70

0.20

0.10

0.15

0.80

0.05

0.15

0.15

0.70�

To Gold

To Galaxy

To Nonsubscriber

PercentChanges

A�1 �15 �2I � A�.

A

A2 � 2A � 5I � O

A�1 �15 �2I � A�.

IA2 � 2A � 5I � O,

A � � 1

�2

2

1�.

�2

1

3

2�� 2

�1

3

�2�

�0

1

1

0��1

0

0

0�A2 � A.

�12,441

16,363

29,372

6,016

12,826

35,288

42,249

73,495

14,231

33,294

8,837

9,957

17,574

3,338

7,085�

�13,048

16,648

25,567

4,935

12,097

33,174

39,486

62,232

11,208

28,037

7,372

8,259

12,438

2,030

4,892�

T.AT

AAT,

A

AAT.AT

T � �1

1

2

4

3

2�A � �0

1

�1

0�

A

T

This collection of thought-provoking and challenging exercises further explores andexpands upon concepts learned in this chapter.

PROBLEM SOLVING

Northeast

Midwest

South

Mountain

Pacific

Northeast

Midwest

South

Mountain

Pacific

638

9. Verify the following equation.



12. Use the equation given in Exercise 11 as a model to find

a determinant that is equal to

13. The atomic masses of three compounds are shown in the

table. Use a linear system and Cramer’s Rule to find the

atomic masses of sulfur (S), nitrogen (N), and fluorine (F).

14. A walkway lighting package includes a transformer, a

certain length of wire, and a certain number of lights on

the wire. The price of each lighting package depends on

the length of wire and the number of lights on the wire.

Use the following information to find the cost of a

transformer, the cost per foot of wire, and the cost of a

light. Assume that the cost of each item is the same in

each lighting package.

• A package that contains a transformer, 25 feet of

wire, and 5 lights costs $20.





15. The transpose of a matrix, denoted is formed by

writing its columns as rows. Find the transpose of each

matrix and verify that

16. Use the inverse of matrix to decode the cryptogram.

23 13 31 63 25 61

24 14 41 20 40

38 116 13 1 22

41 85 28 16

17. A code breaker intercepted the encoded message below.

45 38 18 35 81

42 75 2 22 15

Let

(a) You know that and

that where is the

inverse of the encoding matrix Write and solve

two systems of equations to find and

(b) Decode the message.

18. Let

Use a graphing utility to find Compare with

Make a conjecture about the determinant of the

inverse of a matrix.

19. Let be an matrix each of whose rows adds up to

zero. Find

20. Consider matrices of the form

(a) Write a matrix and a matrix in the form

of

(b) Use a graphing utility to raise each of the matrices

to higher powers. Describe the result.

(c) Use the result of part (b) to make a conjecture about

powers of if is a matrix. Use a graphing

utility to test your conjecture.

(d) Use the results of parts (b) and (c) to make a

conjecture about powers of if is an

matrix.

n nAA

4 4AA

A.

3 32 2

A � �0

0

0

0

0

a12

0

0

0

0

a13

a23

0

0

0

a14

a24

a34

0

0

...

...

...

...

...

...

a1n

a2n

a3n

a�n�1�n

0

� .

A.n nA

A.A�1A�1.

A � �6

0

1

4

2

1

1

3

2�.

z.y,x,w,

A.

A�1�38 �30�A�1 � �8 14�,�45 �35�A�1 � �10 15�

A�1 � �wyx

z�.

�10

�21�2�55�28

�60�30�18�30�35

�32�53

�6�3�11�56

�29�8�17�37

�17�34�34

A � �1

1

1

�2

1

�1

2

�3

4�

A

B � ��3

1

1

0

2

�1�A � ��1

2

1

0

�2

1�,

�AB�T � BTAT.

AT,

ax3 � bx2 � cx � d.

x

�1

0

0

x

�1

c

b

a � ax2 � bx � c

1

a

a3

1

b

b3

1

c

c3 � �a � b��b � c��c � a��a � b � c�

1

a

a2

1

b

b2

1

c

c2 � �a � b��b � c��c � a�

Compound Formula Atomic Mass

Tetrasulfur

tetranitride

Sulfur

hexafluoride

Dinitrogen

tetrafluoride

N2F4

SF6

S4N4 184

146

104

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