8.1 Matrices and Systems of Equations 8.2 Operations with Matrices 8.3...

Matrices can be used to analyzefinancial information such as theprofit a fruit farmer makes on twofruit crops.

SELECTED APPLICATIONSMatrices have many real-life applications. The applications listed below represent a small sample ofthe applications in this chapter.

• Electrical Network,Exercise 82, page 585

• Data Analysis: Snowboarders,Exercise 90, page 585

• Agriculture,Exercise 61, page 599

• Profit,Exercise 67, page 600

• Investment Portfolio,Exercises 67–70, page 609

• Data Analysis: Supreme Court,Exercise 58, page 630

• Long-Distance Plans,Exercise 66, page 634

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

Matrices and Determinants

88D

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etty

Imag

es

571

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MatricesIn this section, you will study a streamlined technique for solving systems oflinear equations. This technique involves the use of a rectangular array of realnumbers called a matrix. The plural of matrix is matrices.

The entry in the ith row and jth column is denoted by the double subscriptnotation For instance, refers to the entry in the second row, third column.A matrix having m 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.

Solutiona. 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 1.

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

3 2.

2 2.

1 4.

1 1.

5

2

7

0

2

40

0

0

0

1 3 0 122

a11, a22, a33, . . .n.m n,m n.n

a23aij.

572 Chapter 8 Matrices and Determinants

What you should learn• Write matrices and identify

their orders.

• Perform elementary row operations on matrices.

• Use matrices and Gaussianelimination to solve systems of linear equations.

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

Why you should learn itYou can use matrices to solvesystems of linear equations in two or more variables. Forinstance, in Exercise 90 on page585, you will use a matrix to find a model for the number ofpeople who participated insnowboarding in the UnitedStates from 1997 to 2001.

Matrices and Systems of Equations8.1

Definition of MatrixIf and are positive integers, an (read “ by ”) matrix is arectangular 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

...

Example 1

The HM mathSpace® CD-ROM andEduspace® for this text contain additional resources related to the concepts discussed in this chapter.

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A matrix derived from a system of linear equations (each written in standardform 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 notincluding the constant terms) is the coefficient matrix of the system.

System:

AugmentedMatrix:

CoefficientMatrix:

Note the use of 0 for the missing coefficient of the -variable in the thirdequation, and also note the fourth column of constant terms in the augmentedmatrix.

When forming either the coefficient matrix or the augmented matrix of asystem, you should begin by vertically aligning the variables in the equations andusing 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?

SolutionBegin by rewriting the linear system and aligning the variables.

Next, use the coefficients and constant terms as the matrix entries. Include zerosfor 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 1is represented by

Now try Exercise 9.

R1.Rn

4 5

R1

R2

R3

R4

1012

31

04

04

53

12

60

............

92

04

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

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

6

Section 8.1 Matrices and Systems of Equations 573

The vertical dots in anaugmented matrix separate thecoefficients of the linear systemfrom the constant terms.

Example 2

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Elementary Row OperationsIn Section 7.3, you studied three operations that can be used on a system of linearequations 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 oper-ations. An elementary row operation on an augmented matrix of a given system oflinear equations produces a new augmented matrix corresponding to a new (butequivalent) system of linear equations. Two matrices are row-equivalent if onecan be obtained from the other by a sequence of elementary row operations.

Although elementary row operations are simple to perform, they involve a lotof arithmetic. Because it is easy to make a mistake, you should get in the habit ofnoting the elementary row operations performed in each step so that you can goback and check your work.

Elementary Row Operations

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.

Now try Exercise 25.

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

Elementary Row Operations1. Interchange two rows.

2. Multiply a row by a nonzero constant.

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

Example 3

Most graphing utilities can per-form elementary row operationson matrices. Consult the user’sguide for your graphing utility forspecific 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.

Techno logy

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In Example 3 in Section 7.3, you used Gaussian elimination with back-substitution to solve a system of linear equations. The next example demonstratesthe matrix version of Gaussian elimination. The two methods are essentially thesame. The basic difference is that with matrices you do not need to keep writingthe variables.

Comparing Linear Systems and Matrix Operations

Linear System Associated Augmented Matrix

Add the first equation to the Add the first row to thesecond 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 thethird 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 21 32 9

y 1

y 32 5

12R3 →

1

0

0

2

1

0

3

3

1

.

...

..

.

..

9

5

2

x 2y 3z 9

y 3z 5

z 2

12 R3.

12

12.

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 R3.22

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 shouldcheck a solution by substitutingthe values of and intoeach equation of the originalsystem. For example, you cancheck the solution to Example 4as follows.

Equation 1:

Equation 2:

Equation 3:21 51 52 17

1 31 4

1 21 32 9

zy,x,

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The last matrix in Example 4 is said to be in row-echelon form. The termechelon refers to the stair-step pattern formed by the nonzero elements of thematrix. To be in this form, a matrix must have the following properties.

Row-Echelon Form

Determine whether each matrix is in row-echelon form. If it is, determinewhether the matrix is in reduced row-echelon form.1

a. b.

c. d.

e. f.

SolutionThe 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 aleading 1 has zeros in every position above and below its leading 1. The matrixin (b) is not in row-echelon form because a row of all zeros does not occur at thebottom of the matrix. The matrix in (e) is not in row-echelon form because thefirst 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


Row-Echelon Form and Reduced Row-Echelon FormA 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 nonzeroentry is 1 (called a leading 1).

3. For two successive (nonzero) rows, the leading 1 in the higher row isfarther to the left than the leading 1 in the lower row.

A matrix in row-echelon form is in reduced row-echelon form if everycolumn that has a leading 1 has zeros in every position above and below itsleading 1.

Example 5

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Gaussian Elimination with Back-SubstitutionGaussian elimination with back-substitution works well for solving systems oflinear equations by hand or with a computer. For this algorithm, the order inwhich the elementary row operations are performed is important. You shouldoperate from left to right by columns, using elementary row operations to obtainzeros in all entries directly below the leading 1’s.

Gaussian Elimination with Back-Substitution

Solve the system

Solution

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

Using back-substitution, 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 fourthcolumns 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 haszeros 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 haszeros 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 sofirst column has leading1 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

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The procedure for using Gaussian elimination with back-substitution issummarized 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 with zeros except for the last entry, it is unnecessary to continue theelimination process. You can simply conclude that the system has no solution, oris 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 of zeros except for the last entry.This means that the original system of linear equations is inconsistent. You cansee 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

.


Gaussian Elimination with Back-Substitution1. 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.

Example 7

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Gauss-Jordan EliminationWith Gaussian elimination, elementary row operations are applied to a matrix toobtain a (row-equivalent) row-echelon form of the matrix. A second method ofelimination, called Gauss-Jordan elimination, after Carl Friedrich Gauss andWilhelm Jordan (1842–1899), continues the reduction process until a reducedrow-echelon form is obtained. This procedure is demonstrated in Example 8.

Gauss-Jordan Elimination

Use Gauss-Jordan elimination to solve the system

SolutionIn Example 4, Gaussian elimination was used to obtain the row-echelon form ofthe linear system above.

Now, apply elementary row operations until you obtain zeros above each of theleading 1’s, as follows.

The matrix is now in reduced row-echelon form. Converting back to a system oflinear equations, you have

Now you can simply read the solution, and which can bewritten as the ordered triple


The elimination procedures described in this section sometimes result infractional 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, whichwill result in working with fractional coefficients. You can sometimes avoidfractions by judiciously choosing the order in which you apply elementary rowoperations.

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 haszeros above its leading 1.

9R3 R1 →3R3 R2 →

1

0

0

0

1

0

0

0

1

.

..

.

..

.

..

1

1

2


so second column has azero 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 asystem of linear equations isthat the solution of the system is easily found without usingback-substitution, as illustratedin Example 8.

For a demonstration of a graphicalapproach to Gauss-Jordan elimi-nation on a matrix, see the Visualizing Row OperationsProgram available for severalmodels of graphing calculators atour website college.hmco.com.

2 3

Techno logy

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Recall from Chapter 7 that when there are fewer equations than variables ina system of equations, then the system has either no solution or infinitely manysolutions.

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 to the system that does not use any of the three variables ofthe 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 representsan infinite number of solutions. Try substituting values for to obtain a few solu-tions. Then check each solution in the original equation.


It is worth noting that the row-echelon form of a matrix is not unique. Thatis, two different sequences of elementary row operations may yield different row-echelon forms. This is demonstrated in Example 10.

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


In Example 9, and are solvedfor in terms of the third variable

To write a solution to thesystem that does not use any ofthe three variables of the system,let represent any real numberand let Then solve for and The solution can then bewritten in terms of which isnot one of the variables of thesystem.

a,y.

xz a.a

z.

yx

Example 9

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Comparing Row-Echelon Forms

Compare the following row-echelon form with the one found in Example 4. Is itthe same? Does it yield the same solution?

SolutionThis row-echelon form is different from that obtained in Example 4. The corre-sponding system of linear equations for this row-echelon matrix is

Using back-substitution on this system, you obtain the solution

and

which is the same solution that was obtained in Example 4.


You have seen that the row-echelon form of a given matrix is not unique;however, the reduced row-echelon form of a given matrix is unique. Try applyingGauss-Jordan elimination to the row-echelon matrix in Example 10 to see thatyou obtain the same reduced row-echelon form as in Example 8.

z 2y 1,x 1,

x 3y

y

4

3z 5

z 2

.

12R3 →

1

0

0

3

1

0

0

3

1

.

..

.

..

.

..

4

5

2

R2 R3 → 1

0

0

3

1

0

0

3

2

.

..

.

..

.

..

4

5

4

R1 R2 →2R1 R3 →

1

0

0

3

1

1

0

3

5

.

..

.

..

.

..

4

5

9

R1 →

1

1

2

3

2

5

0

3

5

.

..

.

..

.

..

4

9

17

R2

R1 1

1

2

3

2

5

0

3

5

.

..

.

..

.

..

4

9

17

1

1

2

2

3

5

3

0

5

.

..

.

..

.

..

9

4

17

x 2yx 3y2x 5y

3z

5z

9

4

17

Example 10

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In Exercises 1– 6, determine the order of the matrix.

1. 2.

3. 4.

5. 6.

In Exercises 7–12, write the augmented matrix for thesystem of linear equations.

7. 8.

9. 10.

11. 12.

In Exercises 13–18, write the system of linear equationsrepresented by the augmented matrix. (Use variables

and if applicable.)

13.

14.

15.

16.

17.

18.

In Exercises 19–22, fill in the blank(s) using elementary rowoperations to form a row-equivalent matrix.

19. 20.

21. 22.

1

0

0

2

2

4

7

3212

1

0

0

1

1

3

4

25

165

1

1

2

1

6

3

4

2

9

1

0

0

1

5

3

4

1

212

41

6

83

4

329

13

2

181

41012

136

1

4

3

83

61

0

4

3

13

4

6

3

8

61

2

4

10

3

5

6

140

20

18

17

101

536

11

257

2321

92

13

121870

35

82

0200

010

410

411

3

508

160

1825

29

206

013

52

0

1272

78

53

02

12

23

74

w,z,y,x,

9x 2y 3z

25y 11z

20

5 7x

19x

5y z 13

8z 10

x

7x

3x

8y

y

5z

15z

8z

8

38

20

x 10y

5x 3y

2x y

2z 2

4z 0

6

7x 4y 22

5x 9y 15 4x

x

3y

3y

5

12

70

65

41 33

94520

3

01

701

1536

037 2

363

5 3 8 7 7 0


Exercises 8.1 The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help.

VOCABULARY CHECK: Fill in the blanks.

1. A rectangular array of real numbers than 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.

9. The process of using row operations to write a matrix in reduced row-echelon form is called ________ ________.

PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com.

a33, . . . , anna22,a11,

333202_0801.qxd 12/5/05 10:59 AM Page 582

www.Eduspace.com


In Exercises 23–26, identify the elementary row opera-tion(s) being performed to obtain the new row-equivalentmatrix.


23.


24.


25.


26.

27. 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

28. 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 29–32, determine whether the matrix is in row-echelon form. If it is, determine if it is also in reducedrow-echelon form.

29.

30.

31.

32.

In Exercises 33–36, write the matrix in row-echelon form.(Remember that the row-echelon form of a matrix is notunique.)

33. 34.

35. 36.

In Exercises 37–42, use the matrix capabilities of a graphingutility to write the matrix in reduced row-echelon form.

37. 38.

39.

40.

41.

42.

In Exercises 43–46, write the system of linear equationsrepresented by the augmented matrix. Then use back-substitution to solve. (Use variables and ifapplicable.)

43. 44.

45.

46. 100

210

211

19

3100

110

21

1

42

21

051

011

02

1

43

z,y,x,

51

15

210

432

31

51

11

124

2

413

32

58

15

210

280

30

11

24

22

48

34

411

59

314

152

3156

29

103

12

304

342

1

34

310

10

012

723

2415

6

14

8

11

18

180

13

2

27

1

153

3148

12

3

11

6

027

510

14

100

010

23

1

1100

200

01

0

431

065

100

300

010

080

100

010

010

050

R4.R3,R1,R2

12.R2

R4.R17

R3.R1

R4.R1

R4.R3

70

34

1241

R1.R22

15.R2

R3.R21

R3.R13

R2.R12

123

21

1

341

1

00

296

378

2114

125

25

4

31

7

27

6

1

00

31

7

75

27

65

270

14

13

5

57

1

563

35

10

45 3

41

34

7

133

01

3982

35

11

8

333202_0801.qxd 12/5/05 10:59 AM Page 583

In Exercises 47–50, an augmented matrix that represents asystem of linear equations (in variables and if appli-cable) has been reduced using Gauss-Jordan elimination.Write the solution represented by the augmented matrix.

47.

48.

49.

50.

In Exercises 51–70, use matrices to solve the system ofequations (if possible). Use Gaussian elimination withback-substitution or Gauss-Jordan elimination.

51. 52.

53. 54.

55. 56.

57. 58.

59. 60.

61. 62.

63. 64.

65. 66.

67.

68.

69. 70.

In Exercises 71–76, use the matrix capabilities of a graphingutility to reduce the augmented matrix corresponding tothe system of equations, and solve the system.

71.

72.

73.

74.

75.

76.

In Exercises 77–80, determine whether the two systems of linear equations yield the same solution. If so, find thesolution using matrices.

77. (a) (b)

78. (a) (b)

79. (a) (b)

80. (a) (b)

81. Use the system

to write two different matrices in row-echelon form thatyield the same solution.

x 3y z 3x 5y 5z 1

2x 6y 3z 8

x y 3z

y 2z

z

15

14

4

x 3y z

y 6z

z

19

18

4

x 6y z 15

y 5z 42

z 8

x 4y 5z

y 7z

z

27

54

8

x 4y

y

3z

z

11

4

2

x 3y 4z

y z

z

11

4

2

x y 2z

y 3z

z

6

8

3

x 2y z

y 5z

z

6

16

3

x 2y

x y

y

z 3w 0

w 0

z 2w 0

x2x3x

y

3y5y

z

z

z

w 0

2w 0

0

x 2y 2z 4w

3x 6y 5z 12w

x 3y 3z 2w

6x y z w

11

30

5

9

2x y3x 4yx 5y

5x 2y

z 2w

w

2z 6w

z w

61

33

2x 10y 2z

x 5y 2z

x 5y z

3x 15y 3z

663

9

3x 3y 12z 6x y 4z 2

2x 5y 20z 10x 2y 8z 4

x 2y 0

x y 6

3x 2y 8

x y

3x 4y

4x 8y

22

4

32

4x 12y 7z 20w 22

3x 9y 5z 28w 30

x 2y z 2w 8

3x 7y 6z 9w 26

2x

4x

8x

3z 3

3y 7z 5

9y 15z 9

x

x

2x

y

y

5z 3

2z 1

z 0

3xx

x

2yyy

z2z4z

1510

14 x

x

2y4yy

3z2zz

280

5

2xx

x

2y3yy

zz

228

14x

2x3x

yy

2y

zzz

142119

2x y

2y

7x 5y

3z 24

z 14

6

x

3x

2x

y

2y

3z

2z

z

2

5

4

x 3y

2x 6y

5

10x 2y

2x 4y

1.5

3

5x 5y

2x 3y

5

72x 6y

x 2y

22

9

x y

2x 4y

4

343x 2y

x 3y

27

13

2x 6y 16

2x 3y 7 x 2y 7

2x y 8

100

010

001

53

0100

010

001

4

104

10

01

610

10

01

34

z,y,x,


333202_0801.qxd 12/5/05 11:00 AM Page 584


82. Electrical Network The currents in an electrical networkare given by the solution of the system

where and are measured in amperes. Solve thesystem of equations using matrices.

83. Partial Fractions Use a system of equations to write thepartial fraction decomposition of the rational expression.Solve the system using matrices.

84. Partial Fractions Use a system of equations to write thepartial fraction decomposition of the rational expression.Solve the system using matrices.

85. Finance A small shoe corporation borrowed $1,500,000to expand its line of shoes. Some of the money wasborrowed at 7%, some at 8%, and some at 10%. Use asystem of equations to determine how much was borrowedat each rate if the annual interest was $130,500 and theamount borrowed at 10% was 4 times the amountborrowed at 7%. Solve the system using matrices.

86. Finance A small software corporation borrowed$500,000 to expand its software line. Some of the moneywas borrowed at 9%, some at 10%, and some at 12%. Usea system of equations to determine how much wasborrowed at each rate if the annual interest was $52,000and the amount borrowed at 10% was times the amountborrowed at 9%. Solve the system using matrices.

In Exercises 87 and 88, use a system of equations to find thespecified equation that passes through the points. Solvethe system using matrices. Use a graphing utility to verifyyour results.

87. Parabola: 88. Parabola:

89. Mathematical Modeling A videotape of the path of aball thrown by a baseball player was analyzed with a gridcovering the TV screen. The tape was paused three times,and the position of the ball was measured each time. Thecoordinates obtained are shown in the table. ( and aremeasured in feet.)

(a) Use a system of equations to find the equation of theparabola that passes through thethree points. Solve the system using matrices.

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

(c) Graphically approximate the maximum height of theball and the point at which the ball struck the ground.

(d) Analytically find the maximum height of the ball andthe point at which the ball struck the ground.

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

y ax 2 bx c

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 ax2 bx cy ax2 bx c

212

8x2

x 12x 1 A

x 1

Bx 1

C

x 12

4x2

x 12x 1

A

x 1

B

x 1

C

x 12

I3I1, I2,

I1 I2

3I1 4I2

I2

I3 0

18

3I3 6

Horizontal distance, x Height, y

0 5.0

15 9.6

30 12.4

90. Data Analysis: Snowboarders The table shows thenumbers of people (in millions) in the United Stateswho participated in snowboarding for selected yearsfrom 1997 to 2001. (Source: National Sporting GoodsAssociation)

(a) Use a system of equations to find the equation ofthe parabola that passes throughthe points. Let represent the year, with corresponding to 1997. Solve the system usingmatrices.

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

(c) Use the equation in part (a) to estimate the numberof people who participated in snowboarding in2003. How does this value compare with the actual2003 value of 6.3 million?

(d) Use the equation in part (a) to estimate in the year2008. Is the estimate reasonable? Explain.

y

t 7ty at2 bt c

y

Model It

Year Number, y

1997 2.8

1999 3.3

2001 5.3

333202_0801.qxd 12/5/05 11:00 AM Page 585

Network Analysis In Exercises 91 and 92, answer thequestions about the specified network. (In a network it isassumed that the total flow into each junction is equal tothe total flow out of each junction.)

91. Water flowing through a network of pipes (in thousands ofcubic meters per hour) is shown in the figure.

(a) Solve this system using matrices for the water flowrepresented by

(b) Find the network flow pattern when and

(c) Find the network flow pattern when and

92. The flow of traffic (in vehicles per hour) through a networkof streets is shown in the figure.

(a) Solve this system using matrices for the traffic flowrepresented by

(b) Find the traffic flow when and

(c) Find the traffic flow when and

Synthesis

True or False? In Exercises 93–95, determine whether thestatement is true or false. Justify your answer.

93. is a matrix.

94. The matrix

is in reduced row-echelon form.

95. The method of Gaussian elimination reduces a matrixuntil a reduced row-echelon form is obtained.

96. Think About It The augmented matrix representsa system of linear equations (in variables and ) thathas been reduced using Gauss-Jordan elimination. Writea system of equations with nonzero coefficients that isrepresented by the reduced matrix. (There are manycorrect answers.)

97. Think About It

(a) Describe the row-echelon form of an augmentedmatrix that corresponds to a system of linearequations that is inconsistent.

(b) Describe the row-echelon form of an augmentedmatrix that corresponds to a system of linearequations that has an infinite number of solutions.

98. Describe the three elementary row operations that can beperformed on an augmented matrix.

99. What is the relationship between the three elementaryrow operations performed on an augmented matrix andthe operations that lead to equivalent systems ofequations?

100. Writing In your own words, describe the differencebetween a matrix in row-echelon form and a matrix inreduced row-echelon form.

Skills Review

In Exercises 101–106, sketch the graph of the function. Donot use a graphing utility.

101.

102.

103.

104.

105.

106. f x 3 ln x

hx lnx 1gx 3x2

f x 2 x1

fx x2 2x 1

x2 1

fx 2x2 4x3x x2

100

010

340

210

zy,x,

0001

0010

0100

04

25

4 2 51

03

26

70

x3 0.x2 150

x3 50.x2 200

xi, i 1, 2, . . . , 5.

350200

x1

x2 x4x3

x5

300 150

x6 0.x5 1000

x7 0.x6 0

xi, i 1, 2, . . . , 7.

600 500

x1

x6 x7

x2

x3 x4 x5

600 500


333202_0801.qxd 12/5/05 11:00 AM Page 586

Section 8.2 Operations with Matrices 587

Equality of MatricesIn Section 8.1, you used matrices to solve systems of linear equations. There is arich mathematical theory of matrices, and its applications are numerous. This section and the next two introduce some fundamentals of matrix theory. It isstandard mathematical convention to represent matrices in any of the followingthree ways.

Two matrices and are equal if they have the same orderand 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.

SolutionBecause two matrices are equal only if their corresponding entries are equal, youcan conclude that

and

Now try Exercise 1.

Be sure you see that for two matrices to be equal, they must have the sameorder and their corresponding entries must be equal. For instance,

but 230

140 2

31

4. 2

4

112 2

21

0.5

a22 0.a21 3,a12 1,a11 2,

a11

a21

a12

a22 2

3

1

0a22a11, a12, a21,

1 ≤ j ≤ n.1 ≤ i ≤ maij bijm nB bijA aij

What you should learn• Decide whether two matrices

are equal.

• Add and subtract matrices andmultiply matrices by scalars.

• Multiply two matrices.

• Use matrix operations tomodel and solve real-life problems.

Why you should learn itMatrix operations can be usedto model and solve real-lifeproblems. For instance, inExercise 70 on page 601, matrixoperations are used to analyzeannual health care costs.

Operations with Matrices

© Royalty-Free/Corbis

8.2

Representation of Matrices1. 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,

Example 1

333202_0802.qxd 12/5/05 10:57 AM Page 587

Matrix Addition and Scalar MultiplicationIn this section, three basic matrix operations will be covered. The first two arematrix addition and scalar multiplication. With matrix addition, you can add twomatrices (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 7(a).

In operations with matrices, numbers are usually referred to as scalars. Inthis text, scalars will always be real numbers. You can multiply a matrix by ascalar by multiplying each entry in by c.Ac

A

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

153

10

21 1

132 1 1

0 12 31 2


Definition of Matrix AdditionIf 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 bijA aij

Definition of Scalar MultiplicationIf is an matrix and is a scalar, the scalar multiple of by

is the matrix given by

cA caij .

m ncAcm nA aij

Example 2

Historical NoteArthur Cayley (1821–1895), aBritish mathematician, inventedmatrices around 1858. Cayleywas a Cambridge Universitygraduate and a lawyer by profession. His groundbreakingwork on matrices was begun as he studied the theory oftransformations. Cayley alsowas instrumental in the devel-opment of determinants.Cayley and two Americanmathematicians, BenjaminPeirce (1809–1880) and his sonCharles S. Peirce (1839–1914),are credited with developing“matrix algebra.”

The

Gra

ng

er C

olle

ctio

n

333202_0802.qxd 12/5/05 10:57 AM Page 588

The symbol represents the negation of which is the scalar productMoreover, 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 additionand 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 Exercises 7(b), (c), and (d).

It is often convenient to rewrite the scalar multiple by factoring out ofevery entry in the matrix. For instance, in the following example, the scalar hasbeen factored out of the matrix.

1252

3212

121125

123

121

121

53

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 12

1

1

0

4

3

0

3

2

6

9

6

6

0

3

12

3

6

32

3332

323031

3431

32 3A 3

2

3

2

2

0

1

4

1

2

B 2

1

1

0

4

3

0

3

2A

2

3

2

2

0

1

4

1

2

3A B.B,3A

A B A 1B.

1B.AA BBA1A.

A,A


Example 3

Consider matrices A, B, and Cbelow. Perform the indicatedoperations and compare theresults.

a. Find and b. Find then add C to

the resulting matrix. Findthen add A to the

resulting matrix.c. Find and then add the

two resulting matrices. Findthen multiply the

resulting matrix by 2.A B,

2B,2A

B C,

A B,B A.A B

C 52

26

B 28

01,A 3

41

7,

Exploration

333202_0802.qxd 12/5/05 10:57 AM Page 589

The properties of matrix addition and scalar multiplication are similar tothose of addition and multiplication of real numbers.

Note that the Associative Property of Matrix Addition allows you to writeexpressions such as without ambiguity because the same sum occursno matter how the matrices are grouped. This same reasoning applies to sums offour 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 thematrix by 3, as follows. Notice that you obtain the same result.

324

01 4

32

7 327

28 6

21624

621

624

612

03 12

9621

324

01 4

32

7 324

01 34

32

7

324

01 4

32

7

1

2

3

1

1

2

0

1

4

2

3

2

2

1

1

A B C


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 the sumof the matrices

and

B 1

2

4

5.

A 2

1

3

0

Techno logy

Properties of Matrix Addition and Scalar MultiplicationLet 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 Propertyc d A cA dA

cA B cA cB

1A A

cd A cdA)

A B C A B C

A B B A

dcm nCB,A,

Example 4

Example 5

333202_0802.qxd 12/5/05 10:57 AM Page 590

One important property of addition of real numbers is that the number 0 isthe additive identity. That is, for any real number For matrices, asimilar property holds. That is, if is an matrix and is the zeromatrix 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 set 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 importantdifferences, which will be discussed later.

Solving a Matrix Equation

Solve for in the equation where

and

SolutionBegin by solving the equation for to obtain

Now, using the matrices and you have

Substitute the matrices.

Subtract matrix from matrix

Multiply the matrix by


13. 4

323

2

23.

B.A 1

34

2

6

2

X 1

33

2

4

1 1

0

2

3B,A

X 1

3B A.

3X B A

X

B 3

2

4

1.A 1

0

2

33X A B,X

X B A x b a

X O B A x 0 b a

X A A B A x a a b a

X A B x a b

m n

2 22 3

O 0

0

0

0O 0

0

0

0

0

02 22 3

m nOA O A.

m nOm nAc.c 0 c


Remember that matrices aredenoted by capital letters. So,when you solve for X, you aresolving for a matrix that makesthe matrix equation true.

Example 6

333202_0802.qxd 12/5/05 10:57 AM Page 591


Matrix MultiplicationThe third basic matrix operation is matrix multiplication. At first glance, thedefinition may seem unusual. You will see later, however, that this definition ofthe product of two matrices has many practical applications.

The definition of matrix multiplication indicates a row-by-column multipli-cation, where the entry in the th row and th column of the product isobtained by multiplying the entries in the th row of by the correspondingentries in the th column of and then adding the results. The general pattern formatrix multiplication is as follows.

Finding the Product of Two Matrices

First, note that the product is defined because the number of columns of isequal to the number of rows of Moreover, the product has order Tofind the entries of the product, multiply each row of by each column of as follows.


9

4

15

1

6

10

13 34

43 2453 04

12 3142 2152 01

AB 1

4

5

3

2

03

4

2

1

B,A3 2.ABB.

AAB

BjAi

ABji

Definition of Matrix MultiplicationIf is an matrix and is an matrix, the product

is an matrix

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

AB cij

m pABn pB bijm 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

Example 7

333202_0802.qxd 12/5/05 10:57 AM Page 592


Be sure you understand that for the product of two matrices to be defined, thenumber of columns of the first matrix must equal the number of rows of thesecond matrix. That is, the middle two indices must be the same. The outside twoindices give the order of the product, as shown below.

Finding the Product of Two Matrices

Find the product where

and

SolutionNote 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

1A

2

1

1

1

3

4

AB

3 13 13 3

10

59

12

3631

21

4

026

2 22 22 2

3

2

4

51

0

0

1 3

2

4

5

53

76

14 00 3124 10 21 12 01 31

22 11 21

AB 12

01

32

21

1

401

2 2.AB3 2.B2 3A

B 2

11

401.A 1

2

0

1

3

2

AB

m pn pm nABBA

EqualOrder of AB

Use the following matrices tofind AB, BA, and What do your results tell youabout matrix multiplication,commutativity, and associativity?

C 3

0

0

1

B 0

2

1

3,

A 1

3

2

4,

ABC.ABC,

Exploration

Example 8

Example 9

333202_0802.qxd 12/5/05 10:57 AM Page 593


Patterns in Matrix Multiplication

a.


In Example 10, note that the two products are different. Even if and are defined, matrix multiplication is not, in general, commutative. That is, formost matrices, This is one way in which the algebra of real numbersand the algebra of matrices differ.

AB BA.

BAAB

1 13 11 3

12

111 2 3

If A is an matrix, the identity matrix has the property that andFor example,

and

IA A1

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 A3

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 MultiplicationLet and be matrices and let be a scalar.

1. Associative Property of Multiplication



4. Associative Property of Scalar MultiplicationcAB cAB AcB

A B)C AC BC

AB C AB AC

ABC ABC

cCB,A,

Definition of Identity MatrixThe matrix that consists of 1’s on its main diagonal and 0’s elsewhereis called the identity matrix of order n and is denoted by

Identity matrix

Note that an identity matrix must be square. When the order is understoodto be you can denote simply by I.Inn,

In 1

0

0...

0

0

1

0...

0

0

0

1...

0

. . .

. . .

. . .

. . .

0

0

0...

1 .

n n

b.

3 31 33 1

2

11

42

2

63

31 2 32

11

Example 10

333202_0802.qxd 12/5/05 10:57 AM Page 594

ApplicationsMatrix multiplication can be used to represent a system of linear equations. Notehow the system

can be written as the matrix equation where is the coefficient matrixof 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 forthe matrix

Solutiona. 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 andand the solution of the matrix equation is


X x1

x2

x3

121.

x3 1,x2 2,x1 1,

I X 100

010

001

...

...

...

121.

A B 102

213

12

2

..

.

..

.

..

.

442

A.B

102

213

12

2 x1

x2

x3

442

AX B,

X.A B

AX B.

x1

2x1

2x2

x2

3x2

x3

2x3

2x3

4

4

2

b1

b2

b3

x1

x2

x3

a11

a21

a31

a12

a22

a32

a13

a23

a33

BXAAX B,

a11x1 a12x2 a13x3 b1

a21x1 a22x2 a23x3 b2

a31x1 a32x2 a33x3 b3


The notation representsthe augmented matrix formedwhen matrix is adjoined tomatrix The notation represents the reduced row-echelon form of the augmentedmatrix that yields the solution tothe system.

I ... XA.

B

A ... B

Example 11

333202_0802.qxd 12/5/05 10:57 AM Page 595

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 tofind the total cost of equipment for each team.

SolutionThe 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 costof equipment for the men’s team is $2448. Notice that you cannot find the totalcost using the product because is not defined. That is, the number ofcolumns of (2 columns) does not equal the number of rows of (1 row).


CEECEC

2130 2448.

8012 645 6015 8015 638 6017

CE 80 6 60124515

153817

C 80 6 60 .

E 12

45

15

15

38

17

CE


Example 12

W RITING ABOUT MATHEMATICS

Problem Posing Write a matrix multiplication application problem that uses thematrix

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

A 20

17

42

30

33

50 .

333202_0802.qxd 12/5/05 10:57 AM Page 596


In Exercises 1–4, find and

1.

2.

3.

4.

In Exercises 5–12, if possible, find (a) (b) (c) and (d)

5.

6.

7.

8.

9.

10.

11.

12.

In Exercises 13–18, evaluate the expression.

13.

14. 61

80 0

35

1 112

71

53

06 7

21

1 1014

86

B 4 6 2A 32

1,

B 84

13A 6

10

430,

B 3

21030

549

21

17142

A 1

350

4

42

48

1

02

16

0,

B 13

14

19

16

07

A 21

21

12

00

11,

B 23

31

42A 2

11

114,

B 1

11

45

10A 62

3

145,

B 34

22A 1

221,

B 21

18A 1

211,

3A 2B.3A,A B,A B,

x 2

17

82y

2

32x

y 2 2x 6

17

818

2

3811

16

30

4132

5154

460

163

0

4132

2x 115

3y 5

43x0

5y

x8 5

12138

x7

2y 4

7222

y.x

Exercises 8.2

VOCABULARY CHECK:

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 Multiplication

(d) (iv) Associative Property of Scalar Multiplication


ABC ABCAB C AB AC

cAB AcBA O A

A B B A

cdA cdAc dA cA dA

A B C A B C

1A A

dcm n,CB,A,

n.n n

333202_0802.qxd 12/5/05 10:57 AM Page 597

www.Eduspace.com

15.

16.

17.

18.

In Exercises 19–22, use the matrix capabilities of a graphingutility to evaluate the expression. Round your results tothree decimal places, if necessary.

19.

20.

21.

22.

In Exercises 23–26, solve fo in the equation, given

and

23. 24.

25. 26.

In Exercises 27–34, if possible, find and state the order ofthe result.

27.

28.

29.

30.

31.

32.

33.

34.

In Exercises 35– 40, use the matrix capabilities of a graph-ing utility to find if possible.

35.

36.

37.

38.

39.

40.

In Exercises 41– 46, if possible, find (a) (b) and (c)(Note: )

41.

42.

43.

44.

45.

46.

In Exercises 47–50, evaluate the expression. Use the matrixcapabilities of a graphing utility to verify your answer.

47. 30

12

12

02

12

04

B 230A 3 2 1 ,

B 1 1 2A 78

1,

B 13

31A 1

11

1,

B 13

31A 3

11

3,

B 03

03A 2

11

4,

B 21

18A 1

422,

A2 AA.A2.BA,AB,

B 78

2216

124A

1548

181222,

B 5240

8535

2760

4582

A 9100

1050

38250

1875,

B 2

732

0.5

01514

1.6A

22113

452

866,

B 3

24168

11510

4

6142110

A 3

125

815

1

691

865,

B 12

515

101216A

11146

1210

2

4129,

B 184

11

2

249A

5210

65

5

315,

AB,

B 6 2 1 6A 1012,

B 680

11160

440A

000

000

53

4,

B 15

0

0

0

18

0

0

012A

500

08

0

007,

B 300

01

0

005A

100

040

00

2,

B 10

27A

140

35

2,

B 2

31

146A

048

10

1

027,

B 14

62A 1

60

1338

217,

B 048

10

1

027A

23

1

146,

AB

2A 4B 2X2X 3A B

2X 2A BX 3A 2B

B [ 02

4

30

1].A [2

13

10

4]

X

1261

2

209

5 14

87

156

0 31

1624

1910

10

3.2111.004

0.055

6.8294.914

3.889 1.630

5.2569.768

3.0908.3354.251

55 1422

1119 22

13206

37

21

54 63

202

14

29

111

3 16

530

14

13 7

96

511

307

32 6

831 24

749

125 2 4 0 14 6 18 9

440

02

13 2

31

62

0


333202_0802.qxd 12/5/05 10:57 AM Page 598


48.

49.

50.

In Exercises 51–58, (a) write the system of linear equationsas a matrix equation, and (b) use Gauss-Jordanelimination on the augmented matrix to solve forthe matrix

51. 52.

53. 54.

55.

56.

57.

58.

59. Manufacturing A corporation has three factories, each ofwhich manufactures acoustic guitars and electric guitars.The number of units of guitars produced at factory in oneday is represented by in the matrix

Find the production levels if production is increased by20%.

60. Manufacturing A corporation has four factories, each ofwhich manufactures sport utility vehicles and pickuptrucks. The number of units of vehicle produced at factory

in one day is represented by in the matrix


61. Agriculture A fruit grower raises two crops, apples andpeaches. Each of these crops is sent to three differentoutlets for sale. These outlets are The Farmer’s Market,The Fruit Stand, and The Fruit Farm. The numbers ofbushels of apples sent to the three outlets are 125, 100, and75, respectively. The numbers of bushels of peaches sent tothe three outlets are 100, 175, and 125, respectively. Theprofit per bushel for apples is $3.50 and the profit perbushel for peaches is $6.00.

(a) Write a matrix that represents the number of bushelsof each crop that are shipped to each outlet Statewhat each entry of the matrix represents.

(b) Write a matrix that represents the profit per bushel ofeach fruit. State what each entry of the matrixrepresents.

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

62. Revenue A manufacturer of electronics produces threemodels of portable CD players, which are shipped to twowarehouses. The number of units of model that areshipped to warehouse is represented by in the matrix

The prices per unit are represented by the matrix

Compute and interpret the result.

63. Inventory A company sells five models of computersthrough three retail outlets. The inventories are representedby

Model

A B C D E

Outlet

The wholesale and retail prices are represented by

Price

Wholesale Retail

Model

Compute and interpret the result.ST

T $840

$1200$1450$2650$3050

$1100$1350$1650$3000$3200

T.

12

3S 3

04

222

231

343

032

S.

BA

B $39.50 $44.50 $56.50.

A 5,0006,0008,000

4,00010,0005,000 .

aijji

BA

bij

B

aij

j.iA

A 10040

9020

7060

3060.

aijji

A 7035

50100

2570 .

aij

j

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

9

6

5

x1 2x2 3x3

x1 3x2 x3

2x1 5x2 5x3

9

6

17

x1 3x2 12 6x1 x2 36

4x1 9x2 13 2x1 3x2 4

x1 4x2 10 2x1 x2 0

2x1 3x2 5 x1 x2 4

X.[A B]

AX B,

3

1575 6 7 1 8 9

04

21

22

40

1

01

2 23

0

35

336

15

21

00

14

33

1

A

B

C

D

E

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64. Voting Preferences The matrix

From

R D I

To

is called a stochastic matrix. Each entry represents the proportion of the voting population thatchanges from party to party and represents the pro-portion that remains loyal to the party from one election tothe next. Compute and interpret

65. Voting Preferences Use a graphing utility to find and for the matrix given in Exercise 64. Can

you detect a pattern as is raised to higher powers?

66. Labor/Wage Requirements A company that manufac-tures boats has the following labor-hour and wage requirements.

Labor per boat

Department

Cutting Assembly Packaging

Boat size

Wages per hour

Plant

A B

Department


67. Profit At a local dairy mart, the numbers of gallons ofskim milk, 2% milk, and whole milk sold over the week-end are represented by

Skim 2% Wholemilk milk milk

The selling prices (in dollars per gallon) and the profits (indollars per gallon) for the three types of milk sold by thedairy mart are represented by

Selling Profitprice

(a) Compute and interpret the result.

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

68. Profit At a convenience store, the numbers of gallons of87-octane, 89-octane, and 93-octane gasoline sold over theweekend are represented by

Octane

87 89 93

The selling prices per gallon and the profits per gallon forthe three grades of gasoline sold by the convenience storeare represents by

Selling Profitprice

Octane

(a) Compute and interpret the result.

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

69. Exercise The numbers of calories burned by individualsof different body weights performing different types ofaerobic exercises for a 20-minute time period are shown inmatrix

Calories burned

120-lb 150-lbperson person

(a) A 120-pound person and a 150-pound person bicycledfor 40 minutes, jogged for 10 minutes, and walked for60 minutes. Organize the time spent exercising in amatrix

(b) Compute and interpret the result.BA

B.

Bicycling

Jogging

Walking

A 10912764

13615979

A.

AB

8789

93

B 1.952.052.15

0.320.360.40

B.

Friday

Saturday

Sunday

A 580560860

840420

1020

320160540

A.

AB

Skim milk

2% milk

Whole milk

B 2.652.853.05

0.650.700.85

B.

Friday

Saturday

Sunday

A 406076

648296

527684

A.

ST

Cutting

Assembly

Packaging

T $12$9$8

$10$8$7

Small

Medium

Large

S 1.0 hr1.6 hr2.5 hr

0.5 hr1.0 hr2.0 hr

0.2 hr0.2 hr1.4 hr

PP8P7,P6,P5,

P 4,P3,

P2.

piij,i

pij i j

R

D

I

P 0.60.20.2

0.10.70.2

0.10.10.8


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Synthesis

True or False? In Exercises 71 and 72, determine whetherthe statement is true or false. Justify your answer.

71. Two matrices can be added only if they have the sameorder.

72.

Think About It In Exercises 73– 80, let matrices andbe of orders and respectively.

Determine whether the matrices are of proper order to per-form the operation(s). If so, give the order of the answer.

73. 74.

75. 76.

77. 78.

79. 80.

81. Think About It If and are real numbers such thatand then However, if and

are nonzero matrices such that then is notnecessarily equal to Illustrate this using the followingmatrices.

82. Think About It If and are real numbers such thatthen or However, if and are

matrices such that it is not necessarily true thator Illustrate this using the following

matrices.

83. Exploration Let and be unequal diagonal matrices ofthe 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 matri-ces. Make a conjecture about a quick rule for such products.

84. Exploration Let and let

and

(a) Find and Identify any similarities with and

(b) Find and identify

Skills Review

In Exercises 85–90, solve the equation.

85.

86.

87.

88.

89.

90.

In Exercises 91–94, solve the system of linear equationsboth graphically and algebraically.

91.

92.

93.

94. 6x 13y 11

9x 5y 41

x 2y

3x y

5

8

8x 3y

6x 7y

17

27

x 4y

5x 8y

9

39

2x3 5x2 12x 30 0

3x3 12x2 5x 20 0

3x 3 22x2 45x 0

4x3 10x2 3x 0

8x2 10x 3 0

3x2 20x 32 0

B2.

i 4.i 3,i 2,A4.A3,A2,

B 0i

i0.A i

00i

i 1

AB

BA

B 11

11A 3

434,

B O.A OAB O,

BAb 0.a 0ab 0,ba

C 2

2

3

3B 1

100,A 0

01

1,

B.AAC BC,

CB,A,a b.ac bc,c 0cb,a,

BC DADA 3BCB DBC D

BCAB

B 3CA 2C

2 2,3 2,2 3,2 3,DC,B,A,

62

26

40

01 4

00

16

226

70. Health Care The health care plans offered this yearby a local manufacturing plant are as follows. Forindividuals, the comprehensive plan costs $694.32, theHMO standard plan costs $451.80, and the HMO Plusplan costs $489.48. For families, the comprehensiveplan costs $1725.36, the HMO standard plan costs$1187.76 and the HMO Plus plan costs $1248.12. Theplant expects the costs of the plans to change next yearas 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 andwhere represents the health care plan costs for

this year and represents the health care plan costsfor next year. State what each entry of each matrixrepresents.

(b) Compute and interpret the result.

(c) The employees receive monthly paychecks fromwhich the health care plan costs are deducted. Usethe matrices from part (a) to write matrices thatshow how much will be deducted from eachemployees’ paycheck this year and next year.

(d) Suppose the costs of each plan instead increase by4% next year. Write a matrix that shows the newmonthly payment.

A B

BAB,

A

Model It

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The Inverse of a MatrixThis section further develops the algebra of matrices. To begin, consider the realnumber equation To solve this equation for multiply each side of theequation by (provided that ).

The number is called the multiplicative inverse of because Thedefinition of the multiplicative inverse of a matrix is similar.

The Inverse of a Matrix

Show that is the inverse of where

and

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

As you can see, This is an example of a square matrix that hasan inverse. Note that not all square matrices have an inverse.

Now try Exercise 1.

Recall that it is not always true that even if both products aredefined. However, if and are both square matrices and it can beshown that So, in Example 1, you need only to check that AB I2.BA In .

AB In ,BAAB BA,

AB I BA.

BA 1

1

2

11

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

1AB I BA,A,B

B 1

1

2

1.A 1

1

2

1A,B

a1a 1.aa1

x a1b

1x a1b

a1ax a1b

ax b

a 0a1x,ax b.


What you should learn• Verify that two matrices are

inverses of each other.

• Use Gauss-Jordan eliminationto find the inverses of matrices.

• Use a formula to find theinverses of matrices.

• Use inverse matrices to solvesystems of linear equations.

Why you should learn itYou can use inverse matrices to model and solve real-lifeproblems. For instance, inExercise 72 on page 610, aninverse matrix is used to finda linear model for the numberof licensed drivers in the UnitedStates.

2 2

The Inverse of a Square Matrix

Jon Love/Getty Images

8.3

Definition of the Inverse of a Square MatrixLet be an matrix and let be the identity matrix. If thereexists a matrix such that

then is called the inverse of The symbol is read “ inverse.”AA1A.A1

AA1 In A1A

A1n nInn nA

Example 1

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Finding Inverse MatricesIf a matrix has an inverse, is called invertible (or nonsingular); otherwise,

is called singular. A nonsquare matrix cannot have an inverse. To see this, notethat 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 allsquare matrices have inverses (see the matrix at the bottom of page 605). If,however, a matrix does have an inverse, that inverse is unique. Example 2 showshow to use a system of equations to find the inverse of a matrix.

Finding the Inverse of a Matrix

Find the inverse of

SolutionTo 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 thatand From the second system you can determine that

and Therefore, the inverse of is

You can use matrix multiplication to check this result.

Check


A1A 3

1

4

11

1

4

3 1

0

0

1

AA1 1

1

4

33

1

4

1 1

0

0

1

3

1

4

1.

X A1

Ax22 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

001

11

43

x11

x21

x12

x22 1

001

X.AX IA,

A 1

1

4

3.

BAABm nn mBm nA

AAA

Section 8.3 The Inverse of a Square Matrix 603

Example 2

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In Example 2, note that the two systems of linear equations have the samecoefficient matrix Rather than solve the two systems represented by

and

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

A I

This “doubly augmented” matrix can be represented as By applyingGauss-Jordan elimination to this matrix, you can solve both systems with a singleelimination 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

1A1

I A1.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

0A.


Finding an Inverse MatrixLet be a square matrix of order

1. Write the matrix that consists of the given matrix on the leftand the identity matrix on the right to obtain

2. If possible, row reduce to using elementary row operations on theentire matrix The result will be the matrix If this isnot possible, is not invertible.

3. Check your work by multiplying to see that AA1 I A1A.

AI A1.A I.

IA

A I.In nAn 2n

n.A

Most graphing utilities can findthe inverse of a square matrix. Todo so, you may have to use theinverse key . Consult the user’sguide for your graphing utility forspecific keystrokes.

Techno logy

x1

333202_0803.qxd 12/5/05 11:01 AM Page 604

Finding the Inverse of a Matrix

Find the inverse of

SolutionBegin 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 usingthis algorithm, if the matrix does not reduce to the identity matrix, then doesnot have an inverse. For instance, the following matrix has no inverse.

To confirm that matrix above has no inverse, adjoin the identity matrix to toform and perform elementary row operations on the matrix. After doingso, you will see that it is impossible to obtain the identity matrix on the left.Therefore, is not invertible.A

IA I

AA

A 1

3

2

2

1

3

0

2

2

AAA.n n

AA1 116

10

2

013

232

334

111

100

010

001 I

I,A1A

A1 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 .

.. A1

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 A1,

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.


Be sure to check your solutionbecause it is easy to makealgebraic errors when usingelementary row operations.

Example 3

333202_0803.qxd 12/8/05 10:44 AM Page 605

The Inverse of a 2 2 MatrixUsing Gauss-Jordan elimination to find the inverse of a matrix works well (evenas a computer technique) for matrices of order or greater. For matrices, however, many people prefer to use a formula for the inverse ratherthan 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 theinverse is given by

Formula for inverse of matrix A

The denominator is called the determinant of the matrix Youwill study determinants in the next section.

Finding the Inverse of a 2 2 Matrix

If possible, find the inverse of each matrix.

a.

b.

Solutiona. 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 theentries on the main diagonal, changing the signs of the other two entries, andmultiplying 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 32 16

B,

14.

1212

1434

d,c,b,a, A1 142

2

1

3

14,

4.

ad bc 32 12

2 2A,

B 3

6

1

2

A 3

2

1

2

A.2 2ad bc

A1 1

ad bc d

c

b

a.

ad bc 0,ad bc 0.A

A a

c

b

d2 2A

2 2

2 23 3


Use a graphing utility withmatrix capabilities to find theinverse of the matrix

What message appears on thescreen? Why does the graphingutility display this message?

A 1

2

3

6.

Exploration

Example 4

333202_0803.qxd 12/5/05 11:01 AM Page 606

Systems of Linear EquationsYou know that a system of linear equations can have exactly one solution,infinitely many solutions, or no solution. If the coefficient matrix of a squaresystem (a system that has the same number of equations as variables) isinvertible, the system has a unique solution, which is defined as follows.

Solving a System Using an Inverse

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% onAAA bonds, 7.5% on AA bonds, and 9.5% on B bonds. You will invest twice asmuch 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.

SolutionBegin 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 to the system is and So, you willinvest $4000 in AAA bonds, $4000 in AA bonds, and $2000 in B bonds.


z 2000.y 4000,x 4000,

400040002000

15 21.5

7.5

200300

100

2 3.5

1.5 10,000

7300

X A1B

A1B

A1 15

21.57.5

200300

100

2 3.5

1.5A1.

xyz

10,000730

010.061

10.0750

1 0.0952

AX B.

zy,x,

x0.06x

x

y0.075y

z

0.095z

2z

10,000730

0

A


A System of Equations with a Unique SolutionIf is an invertible matrix, the system of linear equations represented by

has a unique solution given by

X A1B.

AX BA

Example 5

x1

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

The screen will display thesolution, matrix X.

ENTERBA

X.

BA

Techno logy

333202_0803.qxd 12/5/05 11:01 AM Page 607

In Exercises 1–10, show that B is the inverse of A.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

In Exercises 11–26, find the inverse of the matrix (if itexists).

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

In Exercises 27–38, use the matrix capabilities of a graphingutility to find the inverse of the matrix (if it exists).

27. 28.

29. 30.

31. 32. 56

0

1

1323

12

116

2

52

12

1

0

34

0

1

14

3212

32

4

224

223

13

2

110

203

10

53

512

74

213

5

27

7

11015

1000

3200

24

20

0615

8000

0100

0040

000

5

132

005

005

132

045

005

13

1

27

4

29

7133

156

145

2

60

515

1 23

79

12

21

342

448

111

101

211

74

33191

223

13

272

003

B 1

333

03

11

12

1211

33

00

A 1

11

0

11

11

0121

1001,

B 1

3 44

1

58

2

330A

210

21

1

304,

A 2

12

0

011

1

13

03

00

21

,

A 23

14

001

1

10

21

1110,

B 1214

14

1

1

1

32

11474

A 41

0

12

1

54

1,

B 123

146

235A

21

0

17113

1172,

B 35

25

1515A 1

21

3,

B 232

1

12A 1

3

2

4,

B 21

11A 1

11

2,

B 35

12A 2

513,


Exercises 8.3


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.A1AA1 In A1 A,A1n n

B 14

03

291

5

151

3

161

3

B 312

53

314

64

15

21

210

43

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www.Eduspace.com


33. 34.

35. 36.

37. 38.

In Exercises 39–44, use the formula on page 606 to find theinverse of the matrix (if it exists).

39. 40.

41. 42.

43. 44.

In Exercises 45– 48, use the inverse matrix found in Exercise13 to solve the system of linear equations.

45. 46.

47. 48.

In Exercises 49 and 50, use the inverse matrix found inExercise 21 to solve the system of linear equations.

49. 50.

In Exercises 51 and 52, use the inverse matrix found inExercise 38 to solve the system of linear equations.

51.

52.

In Exercises 53– 60, use an inverse matrix to solve (ifpossible) the system of linear equations.

53. 54.

55. 56.

57. 58.

59. 60.

In Exercises 61–66, use the matrix capabilities of a graphingutility to solve (if possible) the system of linear equations.

61. 62.

63.

64.

65.

66.

Investment Portfolio In Exercises 67–70, consider a personwho 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 investstwice as much in B bonds as in A bonds. Let and represent the amounts invested in AAA, A, and B bonds,respectively.

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

Total Investment Annual Return

67. $10,000 $705

68. $10,000 $760

69. $12,000 $835

70. $500,000 $38,000

x

0.065x

y

0.07y

2y

z

0.09z

z

(total investment)

(annual return)

0

zy,x,

2xx

2xx

5y 4y 2y

w

2z 2w

5z w

3w

117

31

7x

2x4x

x

3yy

y

2w

w

z 2w

w

4113

128

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

8.80.4x

2x

0.8y

4y

1.6

5

18x 12y 13

30x 24y 233x 4y

5x 3y

2

4

x1 2x2 x3 2x4

3x1 5x2 2x3 3x4

2x1 5x2 2x3 5x4

x1 4x2 4x3 11x4

12

03

x1 2x2 x3 2x4

3x1 5x2 2x3 3x4

2x1 5x2 2x3 5x4

x1 4x2 4x3 11x4

01

12

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

1453

9489

7215

3445

125

324

26

3 7

812

552

23

2 2

132

1

255

4

122

4

23511

1020

020

1

10

10

01

01

4203

8526

74

15

146

710

1010

0202

3030

0404

0.60.71

01

0

0.30.2

0.90.1

0.30.5

0.20.20.4

0.30.20.4

333202_0803.qxd 12/5/05 11:01 AM Page 609

71. Circuit Analysis Consider the circuit shown in the figure.The currents and in amperes, are the solution ofthe system of linear equations

where and are voltages. Use the inverse of thecoefficient matrix of this system to find the unknowncurrents for the voltages.

(a)

(b)

Synthesis


73. Multiplication of an invertible matrix and its inverse iscommutative.

74. If you multiply two square matrices and obtain the identitymatrix, you can assume that the matrices are inverses ofone another.

75. If is a matrix then is invertible

if and only if If verify that theinverse is

76. Exploration Consider matrices of the form

(a) Write a matrix and a matrix in the form ofFind the inverse of each.

(b) Use the result of part (a) to make a conjecture about theinverses of matrices in the form of

Skills Review

In Exercises 77 and 78, solve the inequality and sketch thesolution on the real number line.

77. 78.

In Exercises 79– 82, solve the equation. Approximate theresult to three decimal places.

79. 80.

81. 82.

83. Make a Decision To work an extended applicationanalyzing the number of U.S. households with colortelevisions from 1985 to 2005, visit this text’s website atcollege.hmco.com. (Data Source: Nielsen MediaResearch)

ln x lnx 1 0log2 x 2 4.5

2000ex5 4003x2 315

2x 1 < 3x 7 ≥ 2

A.

A.3 32 2

A a11

000

0a22

00

00

a33

0

0000

. . .

. . .

. . .

. . .

. . .

000

ann

.

A1 1

ad bc d

cb

a.

ad bc 0, ad bc 0.

AA ac

bd,2 2A

E1 24 volts, E2 23 volts

E1 14 volts, E2 28 volts

d+

_+

_E2E1

I1

I3

I2

4Ω1Ω

b

c

2Ω

a

E2E1

2I1

I1

4I3

I2 4I3

I2 I3

E1

E2

0

I3,I2,I1,


72. Data Analysis: Licensed Drivers The table showsthe numbers (in millions) of licensed drivers in theUnited States for selected years 1997 to 2001.(Source: U.S. Federal Highway Administration)

(a) Use the technique demonstrated in Exercises57–62 in Section 7.2 to create a system of linearequations for the data. Let represent the year, with

corresponding to 1997.

(b) Use the matrix capabilities of a graphing utility tofind an inverse matrix to solve the system from part(a) and find the least squares regression line

(c) Use the result of part (b) to estimate the number oflicensed drivers in 2003.

(d) The actual number of licensed drivers in 2003 was196.2 million. How does this value compare withyour estimate from part (c)?

y at b.

t 7t

y

Model It

Year Drivers, y

1997 182.7

1999 187.2

2001 191.3

Model It (cont inued)

(e) Use the result of part (b) to estimate when thenumber of licensed drivers will reach 208 million.

333202_0803.qxd 12/5/05 11:01 AM Page 610

Section 8.4 The Determinant of a Square Matrix 611

What you should learn• Find the determinants of

matrices.

• Find minors and cofactors ofsquare matrices.

• Find the determinants ofsquare matrices.

Why you should learn itDeterminants are often used inother branches of mathematics.For instance, Exercises 79–84 onpage 618 show some types ofdeterminants that are usefulwhen changes in variables aremade in calculus.

2 2

The Determinant of a Square Matrix8.4

The Determinant of a 2 2 MatrixEvery 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 nextsection. Historically, the use of determinants arose from special number patternsthat 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 fractionsare the same. This denominator is called the determinant of the coefficient matrixof the system.

Coefficient Matrix Determinant

The determinant of the matrix can also be denoted by vertical bars on both sidesof the matrix, as indicated in the following definition.

In this text, and are used interchangeably to represent thedeterminant of Although vertical bars are also used to denote the absolutevalue of a real number, the context will show which use is intended.

A convenient method for remembering the formula for the determinant of amatrix is shown in the following diagram.

Note that the determinant is the difference of the products of the two diagonals ofthe matrix.

detA a1

a2

b1

b2 a1b2 a2b1

2 2

A.AdetA

A

detA 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 MatrixThe determinant of the matrix

is given by

detA A a1

a2

b1

b2 a1b2 a2b1.

A a1

a2

b1

b2

333202_0804.qxd 12/5/05 11:03 AM Page 611

The Determinant of a 2 2 Matrix

Find the determinant of each matrix.

a.

b.

c.

Solution

a.

b.

c.

Now try Exercise 5.

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 ofthe matrix. For instance, if then detA 2.A 2,

1 1

0 3 3

04 232

detC 02 32

4 4 4 0

22 41

detB 24 1

2 7 4 3

22 13

detA 21 3

2C 0

2

32

4B 2

4

1

2

A 2

1

3

2


Example 1

Use a graphing utility withmatrix capabilities to find thedeterminant of the followingmatrix.

What message appears on thescreen? Why does the graphingutility display this message?

A 1

13

20

2

Exploration

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 result should be 7, as in Example 1(a). Try evaluating the determinants of othermatrices. Consult the user’s guide for your graphing utility for specific keystrokes.

A

A 2

1

3

2

Techno logy

333202_0804.qxd 12/5/05 11:03 AM Page 612


Sign Pattern for Cofactors

matrix

matrix

matrixn n

.

..

.

..

.

..

.

..

.

..

. . .

. . .

. . .

. . .

. . .4 4

3 3

Minors and CofactorsTo define the determinant of a square matrix of order or higher, it isconvenient to introduce the concepts of minors and cofactors.

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

SolutionTo find the minor delete the first row and first column of and evaluate thedeterminant 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 patternof signs for a matrix shown at the upper left.


C33 6C32 3 C31 5

C23 8 C22 4 C21 2

C13 4 C12 5 C11 1

3 3

M33 6M32 3 M31 5

M23 8M22 4 M21 2

M13 4M12 5 M11 1

M12 34 2

1 31 42 50

3

4

2

1

0

1

2

1,

M12,

M11 1

0

2

1 11 02 10

3

4

2

1

0

1

2

1,

AM11,

A 0

3

4

2

1

0

1

2

1.

i ji j

3 3

Minors and Cofactors of a Square MatrixIf is a square matrix, the minor of the entry is the determinantof the matrix obtained by deleting the th row and th column of Thecofactor of the entry is

Ci j 1i jMi j.

ai jCi j

A.jiai jMi jA

Example 2

333202_0804.qxd 12/5/05 11:03 AM Page 613

The Determinant of a Square MatrixThe definition below is called inductive because it uses determinants of matricesof order to define determinants of matrices of order

Try checking that for a matrix

this definition of the determinant yields as previouslydefined.

The Determinant of a Matrix of Order 3 3

Find the determinant of

SolutionNote that this is the same matrix that was in Example 2. There you found thecofactors 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 inthe first row. You could have used any row or column. For instance, you couldhave expanded along the second row to obtain

Second-row expansion

14.

32 14 28

A a21C21 a22C22 a23C23

14.

01 25 14

A a11C11 a12C12 a13C13

C13 4.C12 5,C11 1,

A 0

3

4

2

1

0

1

2

1.

A a1b2 a2b1,

A a1

a2

b1

b2

2 2

n.n 1


Determinant of a Square MatrixIf is a square matrix (of order or greater), the determinant of isthe sum of the entries in any row (or column) of multiplied by theirrespective cofactors. For instance, expanding along the first row yields

Applying this definition to find a determinant is called expanding bycofactors.

A a11C11 a12C12 . . . a1nC1n.

AA2 2A

Example 3

333202_0804.qxd 12/5/05 11:03 AM Page 614

When expanding by cofactors, you do not need to find cofactors of zeroentries, because zero times its cofactor is zero.

So, the row (or column) containing the most zeros is usually the best choice forexpansion by cofactors. This is demonstrated in the next example.

The Determinant of a Matrix of Order 4 4

Find the determinant of

SolutionAfter inspecting this matrix, you can see that three of the entries in the thirdcolumn are zeros. So, you can eliminate some of the work in the expansion byusing the third column.

Because and have zero coefficients, you need only find the cofac-tor To do this, delete the first row and third column of and evaluate thedeterminant 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.

35A 3C13

5.

0 218 317

C13 01314 2

2 2141

3

2

2 3151

3

1

4

1

0

3

1

2

4

2

3

2 C13 1131

0

3

1

2

4

2

3

2AC13.

C43C23, C33,

A 3C13 0C23 0C33 0C43

A 1

1

0

3

2

1

2

4

3

0

0

0

0

2

3

2.

aijCij 0Cij 0


Example 4

333202_0804.qxd 12/5/05 11:03 AM Page 615

In Exercises 1–16, find the determinant of the matrix.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

In Exercises 17–22, use the matrix capabilities of a graphingutility to find the determinant of the matrix.

17. 18.

19. 20.

21. 22.

In Exercises 23–30, find all (a) minors and (b) cofactors ofthe matrix.

23. 24.

25. 26.

27. 28.

29. 30.

In Exercises 31–36, find the determinant of the matrix bythe method of expansion by cofactors. Expand using theindicated row or column.

31. 32.

(a) Row 1 (a) Row 2

(b) Column 2 (b) Column 3

33. 34.

(a) Row 2 (a) Row 3


35. 36.

(a) Row 2 (a) Row 3


In Exercises 37–52, find the determinant of the matrix.Expand by cofactors on the row or column that appears tomake the computations easiest.

37. 38. 2

10

21

1

304

244

122

011

10401

8030

352

3

76

72

64

18

01306

3670

58

42

10300

50

10

5101

501

0126

343

3

64

43

7

21

83

42

25

3

161

2

76

96

7

40

633

1

223

86

6134

12

6

054

43

1

02

1

211

67

52 3

21

4 11

3023

24

5

200

350

122

13

2

461

26

40.17.50.3

0.16.20.6

4.30.7

1.20.9

0.12.2

0.70.34.2

0 1.36.1

0.1

0.30.5

0.20.20.4

0.30.20.4

0.30.2

0.4

0.20.20.4

0.20.20.3

23

1

43

13

12

6

1313

03

629

708

42

753

621

26

392

063

40

307

300

24

23 5

623

35

122

314

85


Exercises 8.4


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

1i j Mij.AaijCij

A.jiaijMij

A.AdetA

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39. 40.

41. 42.

43. 44.

45. 46.

47. 48.

49. 50.

51.

52.

In Exercises 53– 60, use the matrix capabilities of a graph-ing utility to evaluate the determinant.

53. 54.

55. 56.

57. 58.

59.

60.

In Exercises 61– 68, find (a) (b) (c) and(d)

61.

62.

63.

64.

65.

66.

67.

68.

In Exercises 69 –74, evaluate the determinant(s) to verifythe equation.

69.

70.

71.

72.

73.

74.

In Exercises 75–78, solve for

75.

76.

77.

78. x 47

2x 5 0

x 31

2x 2 0

x 23

1x 0

x 13

2x 2 0

x.

a baa

aa b

a

aa

a b b23a b

111 xyz

x 2

y 2

z 2 y xz xz y

wcw

xcx 0

wy xz wy x cw

z cy wy cx

cz cwy xz

wy xz y

wzx

B 203

11

2

431A

213

01

1

120,

B 1

00

020

003A

110

201

110,

B 3

02

02

1

11

1A 3

12

23

0

041,

B 313

21

1

021A

03

0

12

4

211,

B 01

62A 5

34

1,

B 12

12A 4

30

2,

B 10

21A 2

41

2,

B 20

01A 1

003,

AB.AB,B,A,

20000

03000

00

100

00020

0000

4 31

541

20

172

420

83

31300

10202

08

47

3160

81

00

269

141220 1602

8028

44

60

3212

055

007 7

26

052

144

12 5

98

877

041308 8

51

746

50000

21000

04230

03640

22312

3

2163

20000

41025

134

11

52000

1

503

460

2

3201

2105

5400

3621

04

32

61242

3

210

6013

562

1

402

1

2213

6757

6300

2617

3

71

0112

002

200

430

61

5211

140

342

13

1

424

203

1

45

01

1

005

100

230

543

13

2

110

203

604

30

6

703

333202_0804.qxd 12/5/05 11:03 AM Page 617

In Exercises 79–84, evaluate the determinant in which theentries are functions. Determinants of this type occur whenchanges in variables are made in calculus.

79. 80.

81. 82.

83. 84.

Synthesis


85. If a square matrix has an entire row of zeros, the determi-nant will always be zero.

86. If two columns of a square matrix are the same, thedeterminant of the matrix will be zero.

87. Exploration Find square matrices and to demonstratethat

88. Exploration Consider square matrices in which theentries are consecutive integers. An example of such amatrix is

(a) Use a graphing utility to evaluate the determinants offour matrices of this type. Make a conjecture based onthe results.

(b) Verify your conjecture.

89. Writing Write a brief paragraph explaining the differ-ence between a square matrix and its determinant.

90. Think About It If is a matrix of order such thatis it possible to find Explain.

Properties of Determinants In Exercises 91–93, a propertyof determinants is given ( and are square matrices).State how the property has been applied to the givendeterminants and use a graphing utility to verify theresults.

91. If is obtained from by interchanging two rows of orinterchanging two columns of then

(a)

(b)

92. If is obtained from by adding a multiple of a row ofto another row of or by adding a multiple of a column

of to another column of then

(a)

(b)

93. If is obtained from by multiplying a row by a nonzeroconstant or by multiplying a column by a nonzeroconstant then

(a)

(b)

94. Exploration A diagonal matrix is a square matrix withall zero entries above and below its main diagonal.Evaluate the determinant of each diagonal matrix. Make aconjecture based on your results.

(a) (b)

(c)

Skills Review

In Exercises 95–100, find the domain of the function.

95. 96.

97. 98.

99. 100.

In Exercises 101 and 102, sketch the graph of the solutionof the system of inequalities.

101. 102.


103. 104.

105. 106. 6

12

230

02

17

23

24

5

96

25

38

648

11

x y >y ≤

7x 4y ≤

41

10 x

x2x

y ≤≥

y <

83

5

f s 625e0.5sgt lnt 1

Ax 3

36 x2hx 16 x2

gx 3xf x x3 2x

2000 02

00

0010

00031

00

050

00270 0

4

137 812

4

363 12137 2

31

123

52 103 512 2

3B cA.c,

cAB

527 43

6

243 127 10

36

643

15 32 10 3

17B A.A,A

AAAB

12

1

326

402 1

21

623

204

17

6

321

45

2 17

6

45

2

321

B A.A,AAB

BA

2A?A 5,3 3A

47

10

58

11

69

12.

A B A B.BA

x1

x ln x1 ln x x

1ln x1x

ex

exxex

1 xex e2x

2e2x

e3x

3e3x3x 2

13y 2

1 4u1

12v


333202_0804.qxd 12/8/05 10:39 AM Page 618

Section 8.5 Applications of Matrices and Determinants 619

Cramer’s RuleSo far, you have studied three methods for solving a system of linear equations:substitution, elimination with equations, and elimination with matrices. In thissection, you will study one more method, Cramer’s Rule, named after GabrielCramer (1704–1752). This rule uses determinants to write the solution of asystem of linear equations. To see how Cramer’s Rule works, take another lookat the solution described at the beginning of Section 8.4. There, it was pointedout that the system

has a solution

and

provided that Each numerator and denominator in this solutioncan be expressed as a determinant, as follows.

Relative to the original system, the denominator for and is simply the deter-minant of the coefficient matrix of the system. This determinant is denoted by The numerators for and are denoted by and respectively. They areformed by using the column of constants as replacements for the coefficients of

and as follows.

CoefficientMatrix D

For example, given the system

the coefficient matrix, D, and are as follows.

CoefficientMatrix D

24

3838 5

3 24

53 2

45

3DyDx

DyDx,

2x 5y 34x 3y 8

a1

a2

c1

c2c1

c2

b1

b2a1

a2

b1

b2a1

a2

b1

b2

DyDx

y,x

Dy,DxyxD.

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

What you should learn• Use Cramer’s Rule to solve

systems of linear equations.

• Use determinants to find theareas of triangles.

• Use a determinant to test forcollinear points and find anequation of a line passingthrough two points.

• Use matrices to encode anddecode messages.

Why you should learn itYou can use Cramer’s Rule tosolve real-life problems. Forinstance, in Exercise 58 on page630, Cramer’s Rule is used tofind a quadratic model for thenumber of U.S. Supreme Courtcases waiting to be tried.

Applications of Matrices and Determinants

© Lester Lefkowitz /Corbis

8.5

333202_0805.qxd 12/5/05 11:05 AM Page 619

Cramer’s Rule generalizes easily to systems of equations in variables.The value of each variable is given as the quotient of two determinants. Thedenominator is the determinant of the coefficient matrix, and the numerator is the determinant of the matrix formed by replacing the column corresponding tothe variable (being solved for) with the column representing the constants. Forinstance, 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.

SolutionTo 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 1.

y 1.x 2

y Dy

D

43 10

1114

44 30

14

14

14 1

x Dx

D

10

11

2

514

50 22

14

28

14 2

D 43 2

5 20 6 14

4x 2y 10

3x 5y 11

x3 A3A

a11

a21

a31

a12

a22

a32

b1

b2

b3a11

a21

a31

a12

a22

a32

a13

a23

a33a11x1 a12x2 a13x3 b1

a21x1 a22x2 a23x3 b2

a31x1 a32x2 a33x3 b3

x3

nn


Cramer’s RuleIf 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 ofequations. If the determinant of the coefficient matrix is zero, the system haseither no solution or infinitely many solutions.

Aii

xn AnA. . . ,x2

A2A ,x1

A1A ,

A,Ann

Example 1

333202_0805.qxd 12/5/05 11:05 AM Page 620

Using Cramer’s Rule for a 3 3 System

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

SolutionTo 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

Now try Exercise 7.

Remember that Cramer’s Rule does not apply when the determinant of thecoefficient matrix is zero. This would create division by zero, which is undefined.

45

45

24585

345

125

2323

8585

4326

385

245

485

325

?

?

?

1

1

0

0

2

2

45, 3

2, 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

24 0 12

D 213 2

4

3

4 0141

3

3

4 1151

3

2

41

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.


Equation 2 checks.


Equation 3 checks.

333202_0805.qxd 12/5/05 11:05 AM Page 621

Area of a TriangleAnother application of matrices and determinants is finding the area of a trianglewhose 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 shownin Figure 8.1.

SolutionLet and Then, to find the areaof 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

23

Area 1

2 124 0

2

3

1

1

1 11 0 12 3.

11223 1

1 01324 1

1 11424 2

3 x1

x2

x3

y1

y2

y3

1

1

1 124 0

2

3

1

1

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

4, 3,1, 0, 2, 2,


Area of a TriangleThe area of a triangle with vertices and is

where the symbol indicates that the appropriate sign should be chosen toyield a positive area.

±

Area ± 1

2 x1

x2

x3

y1

y2

y3

1

1

1x3, y3x2, y2,x1, y1,

x1 2 3 4

1

2

3

(1, 0)

(2, 2)

(4, 3)

y

FIGURE 8.1

Example 3

Use determinants to find the area of a triangle with vertices and Confirm your answer by plotting the points in a coordinate planeand using the formula

Area 12 baseheight.

7, 5.7, 1,3, 1,

Exploration

333202_0805.qxd 12/5/05 11:05 AM Page 622

Lines in a PlaneWhat if the three points in Example 3 had been on the same line? What wouldhave happened had the area formula been applied to three such points? Theanswer is that the determinant would have been zero. Consider, for instance, thethree collinear points and as shown in Figure 8.2. The areaof 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. (SeeFigure 8.3.)

SolutionLetting and you have

Because the value of this determinant is not zero, you can conclude that the threepoints do not lie on the same line. Moreover, the area of the triangle with verticesat these points is square units.


126 3

6.

24 26 12

21215 1

1 21317 1

1 11417 1

5x1

x2

x3

y1

y2

y3

1

1

1 2

1

7

2

1

5

1

1

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

7, 52, 2, 1, 1,

0.

1

20 12 12

1

2024 1

2

3

1

1

1 1

201223 1

1 11324 1

1 11424 2

34, 3,0, 1, 2, 2,


x1 2 3 4

1

2

3

(0, 1)

(2, 2)

(4, 3)

y

FIGURE 8.2

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

Test for Collinear PointsThree points and are collinear (lie on the same line)if and only if

x1

x2

x3

y1

y2

y3

111 0.

x3, y3x1, y1, x2, y2,

Example 4

333202_0805.qxd 12/5/05 11:05 AM Page 623

The test for collinear points can be adapted to another use. That is, if you aregiven two points on a rectangular coordinate system, you can find an equation ofthe 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 asshown in Figure 8.4.

SolutionLet and Applying the determinant formulafor the equation of a line produces

To evaluate this determinant, you can expand by cofactors along the first row toobtain 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 verticalline through and is

x 2.

4 2x 0

x

2

2

y

0

2

1

1

1 0

2, 22, 0

x 3y 10 0.

x 3y 10 0

x11 y13 1110 0

x1243 1

1 y13 2

1

1

1 114 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


Two-Point Form of the Equation of a LineAn equation of the line passing through the distinct points and

is given by

xx1

x2

yy1

y2

1

1

1 0.

x2, y2x1, y1

x

(2, 4)

(−1, 3)

−1 1 2 3 4

1

2

4

5

y

FIGURE 8.4

Example 5

333202_0805.qxd 12/5/05 11:05 AM Page 624

CryptographyA cryptogram is a message written according to a secret code. (The Greek wordkryptos means “hidden.”) Matrix multiplication can be used to encode anddecode messages. To begin, you need to assign a number to each letter in thealphabet (with 0 assigned to a blank space), as follows.

Then the message is converted to numbers and partitioned into uncoded rowmatrices, 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.

SolutionPartitioning the message (including blank spaces, but ignoring punctuation) intogroups of three produces the following uncoded row matrices.

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


To encode a message, use the techniques demonstrated in Section 8.3 tochoose an invertible matrix such as

and multiply the uncoded row matrices by (on the right) to obtain coded rowmatrices. Here is an example.

Uncoded Matrix Encoding Matrix A Coded Matrix

13 26 211

11

21

1

23

413 5 5

A

A 1

11

21

1

23

4n 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

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 _


Example 6

333202_0805.qxd 12/5/05 11:05 AM Page 625


Encoding a Message

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

SolutionThe coded row matrices are obtained by multiplying each of the uncoded rowmatrices 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.


For those who do not know the encoding matrix decoding the cryptogramfound in Example 7 is difficult. But for an authorized receiver who knows theencoding matrix decoding is simple. The receiver just needs to multiply thecoded row matrices by (on the right) to retrieve the uncoded row matrices.Here is an example.

A1

13 26 211

1

0

10

6

1

8

5

1 13 5 5

A1A,

A,

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

24 23 77.5 20 5618 23 4233 53 1213 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

333202_0805.qxd 12/5/05 11:05 AM Page 626

Decoding a Message

Use the inverse of the matrix

to decode the cryptogram

SolutionFirst find by using the techniques demonstrated in Section 8.3. is thedecoding matrix. Then partition the message into groups of three to form the coded row matrices. Finally, multiply each coded row matrix by (on theright).

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

A1

A1

A1A1

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

W RITING ABOUT MATHEMATICS

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.

Historical NoteDuring World War II, Navajosoldiers 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 expres-sions for important militaryterms, like iron-fish to meansubmarine. Without the NavajoCode Talkers, the Second WorldWar might have had a very different outcome.

© C

orb

is

333202_0805.qxd 12/5/05 11:05 AM Page 627

In Exercises 1–10, use Cramer’s Rule to solve (if possible)the system of equations.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

In Exercises 11–14, use a graphing utility and Cramer’s Ruleto solve (if possible) the system of equations.

11. 12.

13. 14.

In Exercises 15–24, use a determinant and the givenvertices of a triangle to find the area of the triangle.

15. 16.

17. 18.

19. 20.

21.

22.

23.

24.

In Exercises 25 and 26, find a value of such that thetriangle with the given vertices has an area of 4 squareunits.

25.

26.

In Exercises 27 and 28, find a value of such that thetriangle with the given vertices has an area of 6 squareunits.

27.

28. , 3, y5, 31, 0,8, y1, 1,2, 3,

y

4, 2, 3, 5, 1, y5, 1, 0, 2, 2, y

y

2, 4, 1, 5, 3, 23, 5, 2, 6, 3, 50, 2, 1, 4, 3, 52, 4, 2, 3, 1, 5

(6, −1)

(6, 10)

(−4, −5)

−8

4

8

y

x

(4, 3)

( )

1

1

2

3

4

120,

( )52 , 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

x4

(4, 5)

(5, −2)

(0, 0)

−1 4

−2

12345

y

x1 6

x

(0, 0)

(1, 5)

(3, 1)

5

1

2

3

4

5

y

4321

2x 3y 5z 4

3x 5y 9z 7

5x 9y 17z 13

2x y 2z 6

x 2y 3z 0

3x 2y z 6

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

11.510.4x 0.8y 1.6

0.2x 0.3y 2.2

6x 5y

13x 3y

17

763x 2y

6x 4y

2

4

4x 7y

x 6y

47

273x 4y

5x 3y

2

4


Exercises 8.5


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.


AA

x3, y3x2, y2,x1, y1,A

333202_0805.qxd 12/5/05 11:05 AM Page 628

www.Eduspace.com


29. Area of a Region A large region of forest has beeninfested with gypsy moths. The region is roughlytriangular, as shown in the figure. From the northernmostvertex of the region, the distances to the other verticesare 25 miles south and 10 miles east (for vertex ), and 20miles south and 28 miles east (for vertex ). Use a graph-ing utility to approximate the number of square miles inthis region.

30. Area of a Region You own a triangular tract of land, asshown in the figure. To estimate the number of square feetin the tract, you start at one vertex, walk 65 feet east and50 feet north to the second vertex, and then walk 85 feetwest and 30 feet north to the third vertex. Use a graphingutility to determine how many square feet there are in thetract of land.

In Exercises 31–36, use a determinant to determinewhether the points are collinear.

31. 32.

33. 34.

35. 36.

In Exercises 37 and 38, find such that the points arecollinear.

37. 38.

In Exercises 39– 44, use a determinant to find an equationof the line passing through the points.

39. 40.

41. 42.

43. 44.

In Exercises 45 and 46, find the uncoded 1 3 row matricesfor the message. Then encode the message using theencoding matrix.

Message Encoding Matrix

45. TROUBLE IN RIVER CITY

46. PLEASE SEND MONEY

In Exercises 47–50, write a cryptogram for the messageusing the matrix A.

47. CALL AT NOON

48. ICEBERG DEAD AHEAD

49. HAPPY BIRTHDAY

50. OPERATION OVERLOAD

In Exercises 51–54, use to decode the cryptogram.

51.

11 21 64 112 25 50 29 53 23 46 40 75 55 92

52.

136 58 173 72 120 51 95 38 178 73 70 28 242 101 115 4790 36 115 49 199 82

53.

9 38 28 2541 21 31

54.

112 140 83 19 25 13 72 76 61 95 118 71 20 21 38 35 23 36 42 48 32

A 304

42

5

213

4596480199191991

A 11

6

102

01

3

A 57

23

A 13

25

A1

A [ 1

3

1

2

7

4

2

9

7].

4

33

23

2

11

1

11

6

102

01

3

23, 4, 6, 121

2, 3, 52, 1

10, 7, 2, 74, 3, 2, 10, 0, 2, 20, 0, 5, 3

6, 2, 5, y, 3, 52, 5, 4, y, 5, 2

y

2, 3, 3, 3.5, 1, 20, 2, 1, 2.4, 1, 1.60, 1, 4, 2, 2, 522, 1

2, 4, 4, 6, 33, 5, 6, 1, 10, 23, 1, 0, 3, 12, 5

65

85

50

30

N

S

EW

A

C

B

25

2810

20

N

S

EW

CB

A

333202_0805.qxd 12/5/05 11:05 AM Page 629

In Exercises 55 and 56, decode the cryptogram by using theinverse of the matrix A.

55. 20 17 1 62 143 181

56. 13 61 112 106 1124 29 65 144 172

57. The following cryptogram was encoded with a matrix.

8 21 5 10 5 25 5 196 20 40 1 16

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

Synthesis

True or False? In Exercises 59– 61, determine whether thestatement is true or false. Justify your answer.

59. In Cramer’s Rule, the numerator is the determinant of thecoefficient matrix.

60. You cannot use Cramer’s Rule when solving a system oflinear equations if the determinant of the coefficient matrixis zero.

61. In a system of linear equations, if the determinant of thecoefficient matrix is zero, the system has no solution.

62. Writing At this point in the text, you have learned severalmethods for solving systems of linear equations. Brieflydescribe which method(s) you find easiest to use andwhich method(s) you find most difficult to use.

Skills Review

In Exercises 63–66, use any method to solve the system ofequations.

63.

64.

65.

66.

In Exercises 67 and 68, sketch the region determined by theconstraints. Then find the minimum and maximum valuesof the objective function and where they occur, subject tothe constraints.

67. Objective function: 68. Objective function:

Constraints: Constraints:

x 3y ≥ 15 6x y ≤ 40

4x 3y ≥ 24 x 6y ≤ 30

y ≥ 0 y ≥ 0

x ≥ 0 x ≥ 0

z 6x 7yz 6x 4y

5x y z

2x 3y z

4x 10y 5z

7

5

37

x 3y 5z

4x 2y z

5x 3y 2z

14

1

11

3x 8y

2x 12y

11

16

x 7y 22

5x y 26

1818113131015

2 2

1317317599

6525104561215

A [ 1

3

1

2

7

4

2

9

7]


58. Data Analysis: Supreme Court The table shows thenumbers of U.S. Supreme Court cases waiting to betried for the years 2000 through 2002. (Source:Office of the Clerk, Supreme Court of the United States)

(a) Use the technique demonstrated in Exercises 67–70in Section 7.3 to create a system of linear equationsfor 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 frompart (b).

(d) Use the graph from part (c) to estimate when thenumber of U.S. Supreme Court cases waiting to betried will reach 10,000.

y at2 bt c.

t 0t

y

Model It

Year Number of cases, y

2000 8965

2001 9176

2002 9406

333202_0805.qxd 12/5/05 11:05 AM Page 630

Chapter Summary 631

Chapter Summary8

What did you learn?

Section 8.1 Review Exercises Write matrices and identify their orders (p. 572). 1–8

Perform elementary row operations on matrices (p. 574). 9, 10

Use matrices and Gaussian elimination to solve systems of 11–24linear equations (p. 577).

Use matrices and Gauss-Jordan elimination to solve systems of 25–30linear equations (p. 579).

Section 8.2 Decide whether two matrices are equal (p. 587). 31–34

Add and subtract matrices and multiply matrices by scalars (p. 588). 35–48

Multiply two matrices (p. 592). 49–62

Use matrix operations to model and solve real-life problems (p. 595). 63–66

Section 8.3 Verify that two matrices are inverses of each other (p. 602). 67–70

Use Gauss-Jordan elimination to find the inverses of matrices (p. 603). 71–78

Use a formula to find the inverses of matrices (p. 606). 79–82

Use inverse matrices to solve systems of linear equations (p. 607. 83–94

Section 8.4 Find the determinants of matrices (p. 611). 95–98

Find minors and cofactors of square matrices (p. 613). 99–102

Find the determinants of square matrices (p. 614). 103–106

Section 8.5 Use Cramer’s Rule to solve systems of linear equations (p. 619). 107–110

Use determinants to find the areas of triangles (p. 622). 111–114

Use a determinant to test for collinear points and to find an 115–120equation of a line passing through two points (p. 623).

Use matrices to encode and decode messages (p. 625). 121–124

2 2

2 2

333202_080R.qxd 12/5/05 11:08 AM Page 631

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 equationsrepresented 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 equationsrepresented by the augmented matrix. Then use back-substitution to solve the system. (Use variables and )

11.

12.

13.

14.

In Exercises 15–24, use matrices and Gaussian eliminationwith back-substitution to solve the system of equations (ifpossible).

15. 16.

17.

18.

19.

20.

21.

22.

23.

24.

In Exercises 25–28, use matrices and Gauss-Jordan elimina-tion to solve the system of equations.

25.

26.

27.

28. 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

4x2x

2y

3y

4y

3zzz

w 3 0

2w 0 3

2x

3xx

y2y

3y

z3z2zz

w

2w

3w

69

1114

x 2y 6z

2x 5y 15z

3x y 3z

1

4

6

2x y

2x 2y

2x y

2z 4

5

6z 2

2x 3y 3z 3

6x 6y 12z 13

12x 9y z 2

2x 3y z

2x 3y 3z

4x 2y 3z

10

22

2

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

100

810

01

1

27

1100

510

421

134

100

310

91

1

410

2100

210

32

1

920

z.y,x,

43

2

8110

162

12012

122

132

1314

162110

78

4

353

212

1 549

124

702

9103

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 03

32

17

01

64

405

8.1


Review Exercises8

333202_080R.qxd 12/8/05 10:40 AM Page 632

Review Exercises 633

In Exercises 29 and 30, use the matrix capabilities of agraphing utility to reduce the augmented matrix corre-sponding to the system of equations, and solve the system.

29.

30.

In Exercises 31–34, find and

31.

32.

33.

34.

In Exercises 35–38, if possible, find (a) (b) (c) and (d)

35.

36.

37.

38.

In Exercises 39–42, perform the matrix operations. If it isnot possible, explain why.

39.

40.

41.

42.

In Exercises 43 and 44, use the matrix capabilities of agraphing utility to evaluate the expression.

43.

44.

In Exercises 45–48, solve for in the equation given

and

45. 46.

47. 48.

In Exercises 49–52, find if possible.

49.

50.

51.

52.

In Exercises 53–60, perform the matrix operations. If it isnot possible, explain why.

53.

54.

55.

56.

57. 46 6 2

100

320

24

3 400

330

21

21

25

460

62

8

400

12

54

60 6

42

080

156

24

0 64

20

80

B 1

48A 6 5 7,

B 420

1240A

5711

422,

B 4

2015

124030A

5711

422,

B 312

108A 2

32

5,

AB,

2A 5B 3X3X 2A B

6X 4A 3BX 3A 2B

B [ 1

2

4

2

1

4].A [4

13

05

2]X

5 278

02

2 4 46

1

2113

3 81

23

51 6 4

22

73

6

8

20

14

6

8120 5

236

0112

41

8

2156

24

0 8711

124

117

162

191

68

2

0410

71

35 10

14203

B 1

48A 6 5 7,

B 04

20

31240A

5711

422,

B 4

2015

124030A

5711

422,

B 312

108A 2

32

5,

A 3B.4A,A B,A B,

9012x

4

31

x 10

71

52y0

906

431

271

54

0x 3

02

43

y 5

4y26x

5x 1

02

4316

4426

1

x4

05y

18

4

050

1y

x9 1

7129

y.x8.2

4x 12y 2z

x 6y 4z

x 6y z

2x 10y 2z

20128

10

3x y 5z 2w

x 6y 4z w

5x y z 3w

4y z 8w

441

1558

333202_080R.qxd 12/8/05 10:41 AM Page 633

58.

59.

60.

In Exercises 61 and 62, use the matrix capabilities of agraphing utility to find the product.

61.

62.

63. Manufacturing A tire corporation has three factories,each of which manufactures two products. The number ofunits of product produced at factory in one day isrepresented by in the matrix

Find the production levels if production is decreased by5%.

64. Manufacturing A corporation has four factories, each ofwhich manufactures three types of cordless power tools.The number of units of cordless power tools produced atfactory in one day is represented by in the matrix


65. Manufacturing A manufacturing company producesthree kinds of computer games that are shipped to twowarehouses. The number of units of game that areshipped to warehouse is represented by in the matrix

The price per unit is represented by the matrix


66. Long-Distance Plans The charges (in dollars per minute)of two long-distance telephone companies for in-state, state-to-state, and international calls are represented by

Company

A B

Type of call

You plan to use 120 minutes on in-state calls, 80 minuteson state-to-state calls, and 20 minutes on international callseach month.

(a) Write a matrix that represents the times spent on thephone for each type of call.

(b) Compute and interpret the result.

In Exercises 67–70, show that is the inverse of

67.

68.

69.

70.


71. 72.

73. 74.

In Exercises 75–78, use the matrix capabilities of a graphingutility to find the inverse of the matrix (if it exists).

75. 76.

77. 78. 841

1

02

24

2011

82

41

143

1

3442

121

1

662

2

12

1

4318

61

162

12

01

2

311

0

57

22

3

13

41

31

274

297

32

536

554

B 2

3

2

1

1

2

1212

12

A 1

18

10

4

01

2,

B 2

32

334

111A

116

102

013,

B 211

15A 5

1112,

B 27

14A 4

71

2,

A.B8.3

TC

T

In-stateState-to-stateInternational

C 0.070.100.28

0.0950.08 0.25

C.

BA

B $10.25 $14.50 $17.75.

A 820065005400

740098004800.

aijji

A 805090

703060

9080

100

402050.

aijj

A 8040

120100

14080.

aij

ji

24

32

102

15

3

122

4

1112

17

3 32

52

62

314

120

132 1

50

326

10 4

321 2

044

4 2 6 2

02

13


333202_080R.qxd 12/5/05 11:08 AM Page 634

Review Exercises 635

In Exercises 79–82, use the formula below to find theinverse of the matrix, it it exists.

79.

80.

81.

82.

In Exercises 83–90, use an inverse matrix to solve (ifpossible) the system of linear equations.

83.

84.

85.

86.

87.

88.

89.

90.

In Exercises 91–94, use the matrix capabilities of a graphingutility to solve (if possible) the system of linear equations.

91.

92.

93.

94.

In Exercises 95–98, find the determinant of thematrix.

95.

96.

97.

98.

In Exercises 99–102, find all (a) minors and (b) cofactors ofthe matrix.

99.

100.

101.

102.

In Exercises 103–106, find the determinant of the matrix.Expand by cofactors on the row or column that appears tomake the computations easiest.

103.

104.

105.

106. 5

03

1

6146

015

0

0213

3060

0813

418

4

0221

42

5

73

1

14

126

5

403

124

86

4

351

49

2

32

1

258

106

35

64

27

14

1412

2415

5010

305

97

114

82

54

8.4

x 3y 2z

2x 7y 3z

x y 3z

819

3

3x 3y 4z

y z

4x 3y 4z

2

1

1

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

4x 2y

19x 9y

10

47

3x 10y

5x 17y

8

13

5x y

9x 2y

13

24

x 4y

2x 7y

8

5

34

45

52

83

12

310

20

610

743

78

22

A1 1

ad bc[ dc

ba]

333202_080R.qxd 12/5/05 11:08 AM Page 635

In Exercises 107–110, use Cramer’s Rule to solve (ifpossible) the system of equations.

107. 108.

109. 110.

In Exercises 111–114, use a determinant and the given ver-tices of a triangle to find the area of the triangle.

111. 112.

113. 114.

In Exercises 115 and 116, use a determinant to determinewhether the points are collinear.

115.

116.

In Exercises 117–120, use a determinant to find an equa-tion of the line passing through the points.

117. 118.

119. 120.

In Exercises 121 and 122, find the uncoded rowmatrices for the message. Then encode the message usingthe encoding matrix.

Message Encoding Matrix

121. LOOK OUT BELOW

122. RETURN TO BASE

In Exercises 123 and 124, decode the cryptogram by usingthe inverse of the matrix

123. 5 11 2 370 265 225 57 48 33 32 15 20 245 171 147

124. 145 105 92 264 188 160 23 16 15129 84 78 9 8 5 159 118 100 219

152 133 370 265 225 105 84 63

Synthesis

True or False? In Exercises 125 and 126, determinewhether the statement is true or false. Justify your answer.

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

126.

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

128. Writing What is meant by the cofactor of an entry of amatrix? How are cofactors used to find the determinant ofthe matrix?

129. Three people were asked to solve a system of equationsusing an augmented matrix. Each person reduced thematrix to row-echelon form. The reduced matrices were

and

Can all three be right? Explain.

130. Think About It Describe the row-echelon form of anaugmented matrix that corresponds to a system of linearequations that has a unique solution.

131. Solve the equation for

2 3

58 0

.

10

20

30.

10

01

11,

10

21

31,

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 [510

8

476

365].

2

63

16

2

02

1

23

6

202

03

3

1 3

0.7, 3.20.8, 0.2,52, 3, 7

2, 12, 5, 6, 14, 0, 4, 4

0, 5, 2, 6, 8, 11, 7, 3, 9, 3, 15

(4, 2)

4, − 1 2 3

y

12( (

, 1 32( (

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

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


333202_080R.qxd 12/5/05 11:08 AM Page 636

Chapter Test 637

Chapter Test8

Take this test as you would take a test in class. When you are finished, check yourwork against 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 thesystem.

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. Thenencode 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 116

10

2

013

A1 3

6x y 2z

2x 3y z

4x 4y z

410

18 7x 6y

2x 11y

9

49

631

72

5

201

52

8

134659

134

16

6x 4y 10

10x 5y 20

2

24

41

2

6056

104

5

B 44

10A 5

44

4,

AB3A 2B,A,A B,

4x 3y 2z

x y 2z

3x y 4z

14

5

8

1

113

0112

11

13

23

14

165

123

53

3

−2−4

4

6

−24

(−5, 0)

(4, 4)

(3, 2)x

y

2

FIGURE FOR 13

333202_080R.qxd 12/5/05 11:08 AM Page 637

Proofs without words are pictures or diagrams that give a visual understandingof why a theorem or statement is true. They can also provide a starting point forwriting a formal proof. The following proof shows that a determinant is thearea of a parallelogram.

The following is a color-coded version of the proof along with a brief expla-nation 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 greenquadrilateral

Area of Area of white quadrilateral Area of blue Area of yellowArea 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 bd ad bc

(0, d)

(0, 0)

(a, d)

(a, b)

(a, 0)

(a + c, d)

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

ac bd ad bc

(0, d)

(0, 0)

(a, d)

(a, b)

(a, 0)

(a + c, d)

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

2 2

638

Proofs in Mathematics

333202_080R.qxd 12/5/05 11:08 AM Page 638

1. The columns of matrix show the coordinates of thevertices of a triangle. Matrix is a transformation matrix.

(a) Find and Then sketch the original triangle andthe two transformed triangles. What transformation does

represent?

(b) Given the triangle determined by describe thetransformation process that produces the triangle deter-mined by and then the triangle determined by

2. The matrices show the number of people (in thousands) wholived in each region of the United States in 2000 and thenumber of people (in thousands) projected to live in eachregion in 2015. The regional populations are separated intothree age categories. (Source: U.S. Census Bureau)

20000–17 18–64 65 +

20150–17 18–64 65 +

(a) The total population in 2000 was 281,435,000 and theprojected total population in 2015 is 310,133,000.Rewrite the matrices to give the information as percentsof the total population.

(b) Write a matrix that gives the projected change in thepercent of the population in each region and age groupfrom 2000 to 2015.

(c) Based on the result of part (b), which region(s) and agegroup(s) are projected to show relative growth from2000 to 2015?

3. Determine whether the matrix is idempotent. A squarematrix is idempotent if

(a) (b)

(c) (d)

4. Let

(a) Show that where is the identitymatrix 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 cable television to a citywith 100,000 households. Gold Cable Company has 25,000subscribers and Galaxy Cable Company has 30,000 sub-scribers. (The other 45,000 households do not subscribe.)The percent changes in cable subscriptions each year areshown in the matrix below.

Percent Changes

From From From Non-Gold Galaxy subscriber

(a) Find the number of subscribers each company will havein 1 year using matrix multiplication. Explain how youobtained your answer.

(b) Find the number of subscribers each company will havein 2 years using matrix multiplication. Explain how youobtained your answer.

(c) Find the number of subscribers each company will havein 3 years using matrix multiplication. Explain how youobtained your answer.

(d) What is happening to the number of subscribers to eachcompany? What is happening to the number of nonsub-scribers?

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 satisfyingA2 A.

2 2

A 42

x3

x

A 32

x3

x

0.700.200.10

0.150.800.05

0.150.150.70

To Gold

To Galaxy

To Nonsubscriber

PercentChanges

A1 15 2I A.

A

A2 2A 5I O

A1 15 2I A.

IA2 2A 5I O,

A 12

21.

21

32 2

13

20

1101

000

A2 A.

12,58915,88625,9165,226

14,906

34,08141,03868,99812,62633,296

8,16510,10117,4703,2706,565

13,04916,64625,5694,935

12,098

33,17539,48662,23511,21028,036

7,3728,263

12,4372,0314,893

T.AT

AAT,

A

AAT.AT

T 11

24

32A 0

11

0A

T

639

P.S. Problem Solving

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

Northeast

Midwest

South

Mountain

Pacific

Northeast

Midwest

South

Mountain

Pacific

333202_080R.qxd 12/5/05 11:08 AM Page 639

9. Verify the following equation.



12. Use the equation given in Exercise 11 as a model to find adeterminant that is equal to

13. The atomic masses of three compounds are shown in thetable. Use a linear system and Cramer’s Rule to find theatomic masses of sulfur (S), nitrogen (N), and fluorine (F).

14. A walkway lighting package includes a transformer, a cer-tain length of wire, and a certain number of lights on thewire. The price of each lighting package depends on thelength of wire and the number of lights on the wire. Use thefollowing information to find the cost of a transformer, thecost per foot of wire, and the cost of a light. Assume thatthe 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 bywriting its columns as rows. Find the transpose of eachmatrix and verify that

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

23 13 31 63 25 6124 14 41 20 4038 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 thatwhere is the inverse

of the encoding matrix Write and solve two systemsof 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 ofa matrix.

19. Let be an matrix each of whose rows adds up tozero. 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 tohigher powers. Describe the result.

(c) Use the result of part (b) to make a conjecture aboutpowers of if is a matrix. Use a graphingutility to test your conjecture.

(d) Use the results of parts (b) and (c) to make a conjectureabout powers of if is an matrix.n nAA

4 4AA

A.3 32 2

A 000

00

a12

00

00

a13

a23

0

00

a14

a24

a34

00

...

...

...

...

...

...

a1n

a2n

a3n

an1n

0

A.n nA

A.A1A1.

A 601

421

132.

z.y,x,w,A.

A138 30A1 8 14,45 35A1 10 15

A1 wy

xz.

102125528

6030183035

3253631156

2981737173434

A 111

21

1

23

4A

B 3

11

02

1A 12

10

21,

ABT BTAT.

AT,

ax3 bx2 cx d.

x1

0

0x

1

cba ax2 bx c

1aa3

1bb3

1cc3 a bb cc aa b c

1aa2

1bb2

1cc2 a bb cc a

640

N2F4

SF6

S4N4 184

146

104

Compound Formula Atomic mass

Tetrasulphurtetranitride

Sulfurhexafluoride

Dinitrogentetrafluoride

333202_080R.qxd 12/5/05 11:08 AM Page 640

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