CHAPTER 8 Matrices and Determinants - Saddleback College · CHAPTER 8 Matrices and Determinants ......

100
CHAPTER 8 Matrices and Determinants Section 8.1 Matrices and Systems of Equations . . . . . . . . . . . . 719 Section 8.2 Operations with Matrices . . . . . . . . . . . . . . . . . . 737 Section 8.3 The Inverse of a Square Matrix . . . . . . . . . . . . . . 750 Section 8.4 The Determinant of a Square Matrix . . . . . . . . . . . . 765 Section 8.5 Applications of Matrices and Determinants . . . . . . . . 777 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816

Transcript of CHAPTER 8 Matrices and Determinants - Saddleback College · CHAPTER 8 Matrices and Determinants ......

C H A P T E R 8Matrices and Determinants

Section 8.1 Matrices and Systems of Equations . . . . . . . . . . . . 719

Section 8.2 Operations with Matrices . . . . . . . . . . . . . . . . . . 737

Section 8.3 The Inverse of a Square Matrix . . . . . . . . . . . . . . 750

Section 8.4 The Determinant of a Square Matrix . . . . . . . . . . . . 765

Section 8.5 Applications of Matrices and Determinants . . . . . . . . 777

Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790

Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810

Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816

719

C H A P T E R 8Matrices and Determinants

Section 8.1 Matrices and Systems of Equations

■ You should be able to use elementary row operations to produce a row-echelon form (or reduced row-echelon

form) of a matrix.

1. Interchange two rows.

2. Multiply a row by a nonzero constant.

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

■ You should be able to use either Gaussian elimination with back-substitution or Gauss- Jordan elimination tosolve a system of linear equations.

1. Since the matrix has one row andtwo columns, its order is 1 � 2.

2. Since the matrix has one row andfour columns, its order is 1 � 4.

3. Since the matrix has three rows andone column, its order is 3 � 1.

4. Since the matrix has three rows andfour columns, its order is 3 � 4.

5. Since the matrix has two rows andtwo columns, its order is 2 � 2.

6. Since the matrix has two rows andthree columns, its order is 2 � 3.

7.

� 4�1

�33

��

�512�

� 4x

�x

3y

3y

�5

12

8.

�75

4�9

��

2215�

�7x

5x

4y

9y

22

15

9.

�152

10�3

1

�240

���

206�

�x

5x

2x

10y

3y

y

2z

4z

2

0

6

10.

��1�7

3

�80

�1

5�15

8

���

8�38

20��

�x

�7x

3x

8y

y

5z

15z

8z

8

�38

20

11.

� 719

�50

1�8

��

1310�

� 7x

19x

� 5y �

z

8z

13

10

12.

�90

2�25

�311

��

20�5�

�9x � 2y

�25y

3z

11z

20

�5

13.

� x

2x

2y

3y

7

4

�12

2�3

��

74� 14.

�7x

8x

5y

3y

0

�2

�78

�53

��

0�2� 15.

�2x

6x

y

3y

5z

2z

�12

7

2

�206

013

5�2

0

���

�1272�

Vocabulary Check

1. matrix 2. square 3. main diagonal

4. row; column 5. augmented 6. coefficient

7. row-equivalent 8. reduced row-echelon form 9. Gauss-Jordan elimination

720 Chapter 8 Matrices and Determinants

19.

�2R1 � R2 → �10

42

3�1�

�12

410

35�

16.

�4x

�11x

3x

5y

8y

z

6z

18

25

�29

�4

�113

�508

�160

���

1825

�29� 17.

�9x

�2x

x

3x

12y

18y

7y

3z

5z

8z

2z

� 2w

0

10

�4

�10

�9

�213

121870

35

�82

0200

����

010

�4�10

� 18.

�6x

�x

4x

2y

y

8y

z

7z

10z

z

5w

3w

6w

11w

�25

7

23

�21

�6

�140

20

�18

�17

�101

�536

�11

����

�257

23�21

20.

13 R1 →�1

42

�3

83

6�

�34

6�3

86�

21.

15R2 → �1

0

0

1

1

3

4

�25

20

�165

4�

�3R1 � R2 → 2R1 � R3 →

�1

0

0

1

5

3

4

�2

20

�1

6

4�

�1

3

�2

1

8

1

4

10

12

�1

3

6�

23.

Add 5 times Row 2 to Row 1.

��23

5�1

1�8� → �13

30

�1�39�8�

22.

�R1 � R2 →�2R1 � R3 →

�1

0

0

2

�3

2

4

�7

�4

3212

6�

12R1 →

�1

1

2

2

�1

6

4

�3

4

32

2

9

� �

2

1

2

4

�1

6

8

�3

4

3

2

9�

24.

Add 3 times Row 1 to Row 2.

� 3�4

�13

�47� → �3

5�1

0�4�5�

25.

Interchange Row 1 and Row 2. Then add 4 times thenew Row 1 to Row 3.

�0

�14

�13

�5

�5�7

1

563� → �

�100

3�1

7

�7�5

�27

65

27� 26.

Add 2 times Row 1 to Row 2. Add 5 times Row 1 to Row 3.

��1

25

�2�5

4

31

�7

�2�7

6� → ��1

00

�2�9�6

378

�2�11�4�

27.

(a) (b) (c)

(d) (e) This matrix is in reducedrow-echelon form.�

1

0

0

0

1

0

�1

2

0��

1

0

0

2

1

0

3

2

0�

�1

0

0

2

�5

0

3

�10

0��

1

0

0

2

�5

�5

3

�10

�10��

1

0

3

2

�5

1

3

�10

�1�

�1

2

3

2

�1

1

3

�4

�1�

Section 8.1 Matrices and Systems of Equations 721

28.

(a)

(d) �1

0

0

0

5

2

19

�34�

�7

0

�3

1

1

2

4

5�

�7

0

�3

4

1

2

4

1�

(b)

(e) �1

0

0

0

5

1

19

�34�

�1

0

�3

7

5

2

4

1� (c)

(f) �1

0

0

0

0

1

0

0�

�1

0

0

7

5

2

19

1�

This matrix is in reduced row-echelon form.

29.

This matrix is in reduced row-echelon form.

�100

010

010

050� 30.

This matrix is in reduced row-echelon form.

�100

300

010

080�

31.

The first nonzero entries in Rows 1 and 2 are not 1. The matrix is not in row-echelon form.

�200

0�1

0

431

065� 32.

This matrix is in row-echelon form.

�100

010

2�3

1

1100�

33.

�3R2 � R3 → �1

0

0

1

1

0

0

2

1

5

0

�1�

2R1 � R2 → �3R1 � R3 → �

1

0

0

1

1

3

0

2

7

5

0

�1�

�1

�2

3

1

�1

6

0

2

7

5

�10

14�

35.

�2R2 � R3→ �1

0

0

�1

1

0

�1

6

0

1

3

0�

�5R1 � R2 →

6R1 � R3 → �1

0

0

�1

1

2

�1

6

12

1

3

6�

�1

5

�6

�1

�4

8

�1

1

18

1

8

0�

34.

�3R2 � R3 → �

1

0

0

2

1

0

�1

�2

1

3

5

�1�

�3R1 � R2 →2R1 � R3 →

�1

0

0

2

1

3

�1

�2

�5

3

5

14�

�1

3

�2

2

7

�1

�1

�5

�3

3

14

8�

36.

�2R2 � R3 → �

1

0

0

�3

1

0

0

1

0

�7

2

0�

3R1 � R2 →�4R1 � R3 →

�1

0

0

�3

1

2

0

1

2

�7

2

4�

�1

�3

4

�3

10

�10

0

1

2

�7

23

�24�

37. Use the reduced row-echelon form feature of a graphing utility.

�3

�12

304

3�4�2� ⇒ �

100

010

001�

38. Use the reduced row-echelon form feature of a graphing utility.

�152

3156

29

10� ⇒ �100

300

010�

722 Chapter 8 Matrices and Determinants

39. Use the reduced row-echelon form featureof a graphing utility.

�11

�24

22

�48

34

�411

�5�9

3�14

� ⇒ �1000

2000

0100

0010�

40. Use the reduced row-echelon form feature of a graphing utility.

��2

413

3�2

58

�15

�2�10

�280

�30� ⇒ �

1000

0100

0010

0001�

41. Use the reduced row-echelon form featureof a graphing utility.

��31

5�1

11

124� ⇒ �1

001

32

1612�

42. Use the reduced row-echelon form feature of a graphing utility.

� 5�1

15

210

4�32� ⇒ �1

001

02

2�6�

43.

Solution: ��2, �3�

x � �2

x � 2��3� � 4

�x � 2y

y

4

�3

44.

Solution: �5, �1�

x � 5

x � 5��1� � 0

�x � 5y

y

0

�1

45.

Solution: �8, 0, �2�

x � 8

x � 0 � 2��2� � 4

y � 0

y � ��2� � 2

�x � y

y

2z

z

z

4

2

�2

46.

Solution: ��31, 12, �3�

x � �31

x � 2�12� � 2��3� � �1

y � 12

y � ��3� � 9

z � �3

y � z � 9

x � 2y � 2z � �1

47.

Solution: �3, �4�

y � �4

x � 3

�1

0

0

1��

3

�4� 48.

Solution: ��6, 10�

x

y

�6

10

�10

01

��

�610�

49.

Solution: ��4, �10, 4�

z � 4

y � �10

x � �4

�1

0

0

0

1

0

0

0

1

���

�4

�10

4� 50.

Solution: �5, �3, 0�

x

y

z

5

�3

0

�100

010

001

���

5�3

0� 51.

Solution: �3, 2�

x � 2�2� � 7 ⇒ x � 3

y � 2

�x � 2y

y

7

2

�13R2 →

�10

21

��

72�

�2R1 � R2 → �1

02

�3��

7�6�

�12

21

��

78�

� x

2x

2y

y

7

8

Section 8.1 Matrices and Systems of Equations 723

52.

Solution: ��1, 3�

x � 3�3� � 8 ⇒ x � �1

y � 3

�x � 3y

y

8

3

12R1 →

�13R2 →

�1

0

3

1

��

8

3�

�R1 � R2 →

�20

6�3

��

16�9�

�22

63

��

167�

�2x

2x

6y

3y

16

7

54.

Solution: �9, 13�

x � 13 � �4 ⇒ x � 9

y � 13

�x � y

y

�4

13

��1�R2 → �1

0

�1

1��

�4

13�

�R1 � R2 → �1

0

�1

�1��

�4

�13�

��1�R1 →

�12�R2 →

�1

1

�1

�2

��

�4

�17�

��12

1�4

��

4�34�

��x

2x

y

4y

4

�34

55.

Solution: ��1, �4�

x � 2��4� � �9 ⇒ x � �1

y � �4

�x � 2y

y

�9

�4

110R2 →

�10

21

��

�9�4�

2R1 � R2 → �10

210

��

�9�40�

R1

R2 � 1

�226

��

�9�22�

��21

62

��

�22�9�

��2x

x

6y

2y

�22

�9

56.

Solution: ��2, �1�

x � ��1� � �1 ⇒ x � �2

y � �1

�x � y

y

�1

�1

�15R2 →

�10

�11

��

�1�1�

2R1 � R2 → �1

0�1�5

��

�15�

15R1 → � 1

�2�1�3

��

�17�

� 5�2

�5�3

��

�57�

� 5x

�2x

5y

3y

�5

757.

The system is inconsistent and there is no solution.

2R1 � R2 → ��1

020

��

1.56.0�

��12

2�4

��

1.53.0�

��x

2x

2y

4y

1.5

3.0

53.

Solution: ��5, 6�

x � 3�6� � 13 ⇒ x � �5

y � 6

�x � 3y

y

13

6

�111R2 →

�10

31

��

136�

�3R1 � R2 → �1

03

�11��

13�66�

R1

R2 �1

33

�2��

13�27�

�31

�23

��

�2713�

�3x

x

2y

3y

�27

13

724 Chapter 8 Matrices and Determinants

58.

Solution: where a is a real number�3a � 5, a�

x � 3a � 5

y � a

x � 3y � 5

2R1 � R2 →

�1

0

�3

0��

5

0�

� 1

�2

�3

6��

5

�10�

� x

�2x

3y

6y

5

�1059.

Solution: �4, �3, 2�

x � 3�2� � �2 ⇒ x � 4

y � 7�2� � 11 ⇒ y � �3

z � 2

�x

y

3z

7z

z

�2

11

2

�17R3 → �

100

010

�371

���

�2112�

�2R2 � R3 → �1

0

0

0

1

0

�3

7

�7

���

�2

11

�14�

�3R1 � R2 → �2R1 � R3 → �

1

0

0

0

1

2

�3

7

7

���

�2

11

8�

�1

3

2

0

1

2

�3

�2

1

���

�2

5

4�

�x

3x

2x

y

2y

3z

2z

z

�2

5

4

60.

�R2 →� 1

540R3 → �

1

0

0

�2

1

0

�9

67

1

���

�66

412

6�

9R2 � R3 → �

1

0

0

�2

�1

0

�9

�67

�540

���

�66

�412

�3240�

�R3 � R2 → �1

0

0

�2

�1

9

�9

�67

63

���

�66

�412

468�

4R2 → �1

0

0

�2

8

9

�9

�4

63

���

�66

56

468�

�7R1 � R3 → �

1

0

0

�2

2

9

�9

�1

63

���

�66

14

468�

R3 � ��3�R1 → �

1

0

7

�2

2

�5

�9

�1

0

���

�66

14

6�

�2

0

7

�1

2

�5

3

�1

0

���

24

14

6�

�2x

7x

y

2y

5y

3z

z

24

14

6

Solution: �8, 10, 6�

x � 2�10� � 9�6� � �66 ⇒ x � 8

y � 67�6� � 412 ⇒ y � 10

z � 6

�x � 2y

y

9z

67z

z

�66

412

6

Section 8.1 Matrices and Systems of Equations 725

61.

Solution: �7, �3, 4�

x � ��3� � 4 � 14 ⇒ x � 7

y � 4 � �7 ⇒ y � �3

z � 4

�x � y

y

z

z

z

14

�7

4

13R3 →

�100

�110

1�1

1

���

14�7

4�

�5R2 � R3 → �100

�110

1�1

3

���

14�712�

�2R1 � R2 →�3R1 � R3 →

�100

�115

1�1�2

���

14�7

�23�

�R1 →

�123

�1�1

2

111

���

142119�

��1

23

1�1

2

�111

���

�142119�

��x

2x

3x

y

y

2y

z

z

z

�14

21

19

63.

�x � 2y

y

3z12z

z

�28

0

6

�211R3 → �

100

210

�312

1

���

�2806�

�3R2 � R3 → �100

210

�312

�112

�28

0

�33�

14R2 →

R1 � R3 → �100

213

�312

�4

���

�280

�33�

�10

�1

241

�32

�1

���

�280

�5��

x

�x

2y

4y

y

3z

2z

z

�28

0

�5

62.

Solution: ��6, 8, 2�

x � 3�8� � 2 � �28 ⇒ x � �6

y �12�2� � 7 ⇒ y � 8

z � 2

�x � 3y

y

z12z

z

�28

7

2

�12R2 → �

100

�310

1�

12

1

���

�2872�

4R2 � R3 → �

100

�3�2

0

111

���

�28�14

2�

R1 � R2 →�2R1 � R3 →

�100

�3�2

8

11

�3

���

�28�14

58�

R3

R2

�1

�12

�312

10

�1

���

�28142�

R2

R1 �12

�1

�321

1�1

0

���

�282

14�

�21

�1

2�3

1

�110

���

2�28

14��

2x

x

�x

2y

3y

y

z

z

2

�28

14

Solution: ��4, �3, 6�

x � 2��3� � 3�6� � �28 ⇒ x � �4

y �12�6� � 0 ⇒ y � �3

z � 6

726 Chapter 8 Matrices and Determinants

65.

Solution: where a is any real number.�2a � 1, 3a � 2, a�

x � 2a � 1 ⇒ x � 2a � 1

y � 3a � 2 ⇒ y � 3a � 2

Let z � a.

�x

y

2z

3z

1

2

�R2 � R1 →

3R2 � R3 → �100

010

�2�3

0

���

120�

�R2 → �1

0

0

1

1

�3

�5

�3

9

���

3

2

�6�

�R1 � R2 →

�2R1 � R3 → �1

0

0

1

�1

�3

�5

3

9

���

3

�2

�6�

�1

1

2

1

0

�1

�5

�2

�1

���

3

1

0�

�x

x

2x

y

y

5z

2z

z

3

1

0

66.

Solution: where a is a real number�� 32a �

32, 13a �

13, a�

x � � 32a �

32

y �13a �

13

z � a

12R1 →

�13R2 → �

1

0

0

0

1

0

32

� 13

0

���

3213

0 �

�3R2 � R3 → �

2

0

0

0

�3

0

3

1

0

���

3

�1

0�

�2R1 � R2 →�4R1 � R3 →

�2

0

0

0

�3

�9

3

1

3

���

3

�1

�3�

�2

4

8

0

�3

�9

3

7

15

���

3

5

9�

�2x

4x

8x

3y

9y

3z

7z

15z

3

5

9

67.

Solution:where a and b are real numbers�4 � 5b � 4a, 2 � 3b � 3a, b, a�

x � 5b � 4a � 4 ⇒ x � 4 � 5b � 4a

y � 3b � 3a � 2 ⇒ y � 2 � 3b � 3a

Let w � a and z � b.

�x

y

5z

3z

4w

3w

4

2

�2R2 � R1 → �1

001

�53

�43

��

42�

�3R1 � R2 → �10

21

13

23

��

82�

�13

27

16

29

��

826�

� x

3x

2y

7y

z

6z

2w

9w

8

26

64.

Solution: �5, �1, �2�

x � ��1� � 4��2� � 14 ⇒ x � 5

y � 13��2� � �27 ⇒ y � �1

z � �2

�x � y

y

4z

13z

z

14

�27

�2

12R3 →�

100

�110

�4131

���

14�27�2�

R3

R2

�100

�110

�413

�2

���

14�27

4�

R1 � R2 →�3R1 � R3 →

�100

�101

�4�213

���

144

�27�

R3

R1

�1

�13

�11

�2

�421

���

14�10

15�

�3

�11

�21

�1

12

�4

���

15�10

14��

3x

�x

x

2y

y

y

z

2z

4z

15

�10

14

Section 8.1 Matrices and Systems of Equations 727

68.

Solution:where a and b are real numbers��3b � 96a � 100, b, 52a � 54, a�

x � �3b � 96a � 100

y � b

z � 52a � 54

w � a

2R2 � R1 → �1

0

3

0

0

1

�96

�52��

100

54�

�3R1 � R2 →

�1

0

3

0

�2

1

8

�52��

�8

54�

�R2 � R1 → �1

3

3

9

�2

�5

8

�28��

�8

30�

�4

3

12

9

�7

�5

�20

�28��

22

30�

�4x

3x

12y

9y

7z

5z

20w

28w

22

30

69.

The system is inconsistent and there is no solution.

�R2 � R3 → �1

0

0

�1

1

0

���

22

�627

1607�

17R2 →

�14R3 →�

100

�111

���

22�

627

14�

�3R1 � R2 →�4R1 � R3 → �

100

�17

�4

���

22�62�56�

�R1 →

�134

�14

�8

���

224

32�

��1

34

14

�8

���

�224

32��

�x

3x

4x

y

4y

8y

�22

4

32

70.

The system in inconsistent and there is no solution.

�8R2 � R3 → �

1

0

0

2

�1

0

0

6

�40�

�R1 � R2 →�3R1 � R3 →

�1

0

0

2

�1

�8

0

6

8�

�1

1

3

2

1

�2

���

0

6

8�

�x

x

3x

2y

y

2y

0

6

8

71. Use the reduced row-echelon form feature of a graphing utility.

Let

Solution: where a is any real number�0, 2 � 4a, a�

x � 0

y � 2 � 4a

z � a.

⇒ � x

y � 4z

0

2�312

�1

3152

124

208

����

62

104� ⇒ �

1000

0100

0400

����

0200��

3x

x

2x

�x

3y

y

5y

2y

12z

4z

20z

8z

6

2

10

4

728 Chapter 8 Matrices and Determinants

72. Use the reduced row-echelon form feature of a graphing utility.

Solution: where a is a real number��5a, a, 3�

x � 5a � 0 ⇒ x � �5a

y � a

z � 3

� z

x � 5y�

3

0

�211

�3

1055

�15

221

�3

����

663

�9� ⇒ �

1000

5000

0100

����

0300�

�2x

x

x

�3x

10y

5y

5y

15y

2z

2z

z

3z

6

6

3

�9

73. Use the reduced row-echelon form feature of a graphing utility.

Solution: �1, 0, 4, �2�

w � �2

z � 4

y � 0

x � 1

�2

3

1

5

1

4

5

2

�1

0

2

�1

2

1

6

�1

����

�6

1

�3

3� ⇒ �

1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1

����

1

0

4

�2��

2x

3x

x

5x

y

4y

5y

2y

z

2z

z

2w

w

6w

w

�6

1

�3

3

74. Use the reduced row-echelon form feature of a graphing utility.

Solution: ��1, 1, 3, 1�

x � �1

y � 1

z � 3

w � 1

�x

yz

w

�1131

�1316

263

�1

25

�3�1

41221

����

1130

�5�9

� ⇒ �1000

0100

0010

0001

����

�1131�

�x

3x

x

6x

2y

6y

3y

y

2z

5z

3z

z

4w

12w

2w

w

11

30

�5

�9

Section 8.1 Matrices and Systems of Equations 729

75. Use the reduced row-echelon form feature of a graphing utility.

Let Then and

Solution: where a is a real number��2a, a, a, 0�

y � a.x � �2az � a.

�x

y

2z

z

w

0

0

0

�123

135

111

1�2

0

���

000� ⇒ �

100

010

2�1

0

001

���

000��

x

2x

3x

y

3y

5y

z

z

z

w

2w

0

0

0

76.

Solution: where a is a real number��2a, �a, a, a�

x � �2ay � �a,z � a,w � a,

�x

y

z

2w

w

w

0

0

0

�110

2�1

1

10

�1

312

���

000� ⇒ �

100

010

001

21

�1

���

000�

�x

x

2y

y

y

z

z

3w

w

2w

0

0

0

77. (a)

Solution:

Both systems yield the same solution, namely ��1, 1, �3�.

��1, 1, �3�

x � �1

x � 2�1� � ��3� � �6

y � 1

y � 5��3� � 16

�x � 2y

y

z

5z

z

�6

16

�3

(b)

Solution: ��1, 1, �3�

x � �1

x � �1� � 2��3� � 6

y � 1

y � 3��3� � �8

�x � y

y

2z

3z

z

6

�8

�3

78. (a)

The systems do not yield the same solution.

x � �25

x � 3��2� � 4�2� � �11

y � �2

y � 2 � �4

�x � 3y

y

4z

z

z

�11

�4

2

(b)

x � �3

x � 4��2� � �11

y � �2

y � 3�2� � 4

�x � 4y

y � 3z

z

�11

4

2

730 Chapter 8 Matrices and Determinants

79. (a)

Solution:

The systems do not yield the same solution.

��5, 2, 8�

x � �5

x � 4�2� � 5�8� � 27

y � 2

y � 7�8� � �54

�x � 4y

y

5z

7z

z

27

�54

8

(b)

Solution: �19, 2, 8�

x � 19

x � 6�2� � �8� � 15

y � 2

y � 5�8� � 42

�x � 6y

y

z

5z

z

15

42

8

80. (a)

The systems do not yield the same solution.

x � �3

x � 3�6� � ��4� � 19

y � 6

y � 6��4� � �18

�x � 3y

y

z

6z

z

19

�18

�4

(b)

x � 3

x � 6 � 3��4� � �15

y � 6

y � 2��4� � 14

�x � y

y

3z

2z

z

�15

14

�4

81.

There are infinitely many matrices in row-echelon form that correspond to the original system of equations. All suchmatrices will yield the same solution, namely �16, �5, 2�.

The row-echelonform feature of agraphing utilityyields this form.

�1

00

3

10

3274

1

��

4

�32

2�

This is a matrixin row-echelonform.

12R2 →

100

310

121

���

3�1

2�

�R1 � R2 →�2R1 � R3 →

�100

320

141

���

3�2

2�

�112

356

153

���

318�

�x

x

2x

3y

5y

6y

z

5z

3z

3

1

8

82.

I1 � 3 � 1 � 0 ⇒ I1 � 2

I2 � 3�1� � 6 ⇒ I2 � 3

I3 � 1

�I1 � I2

I2

I3

3I3

I3

0

6

1

�1

24R3 → �

100

�110

131

���

061�

�7R2 � R3 → �

100

�110

13

�24

���

06

�24�

R3

R2

�100

�117

13

�3

���

06

18�

�3R1 � R2 → �100

�171

1�3

3

���

0186�

�130

�141

103

���

0186�

�I1

3I1

I2

4I2

I2

I3

3I3

0

18

6

Section 8.1 Matrices and Systems of Equations 731

84.

System of equations:

8x2

�x � 1�2�x � 1� �2

x � 1�

6x � 1

�4

�x � 1�2

A � 2, B � 6, C � 4

�1

�21

10

�1

011

���

800� rref

→ �100

010

001

���

264�

A�2A

A

B

B�

CC

800

8x2 � �A � B�x2 � ��2A � C�x � �A � B � C�

8x2 � A�x2 � 2x � 1� � B�x2 � 1� � C�x � 1�

8x2 � A�x � 1�2 � B�x � 1��x � 1� � C�x � 1�

8x2

�x � 1�2�x � 1� �A

x � 1�

Bx � 1

�C

�x � 1�2

83.

System of equations:

Thus,

So,4x2

�x � 1�2�x � 1� �1

x � 1�

3x � 1

�2

�x � 1�2.

A � 1, B � 3, C � �2.

�100

010

001

���

13

�2�→rref�

121

10

�1

01

�1

���

400� A � B � C � 0

2A � C � 0

A � B � 4

4x2 � �A � B�x2 � �2A � C�x � �A � B � C�

4x2 � A�x2 � 2x � 1� � B�x2 � 1� � C�x � 1�

4x2 � A�x � 1�2 � B�x � 1��x � 1� � C�x � 1�

4x2

�x � 1�2�x � 1��

A

x � 1�

B

x � 1�

C

�x � 1�2

85. amount at 7%

amount at 8%,

amount at 10%

Solution: $150,000 at 7%, $750,000 at 8%,and $600,000 at 10%

x � 750,000 � 600,000 � 1,500,000 ⇒ x � 150,000

y � 3�600,000� � 2,550,000 ⇒ y � 750,000

�x � y

y

z

3z

z

1,500,000

2,550,000

600,000

�17R3 →

�100

110

131

���

1,500,0002,550,000

600,000�

�4R2 � R3 → �

100

110

13

�7

���

1,500,0002,550,000

�4,200,000�

100R2 → �

100

114

135

���

1,500,0002,550,0006,000,000�

�0.07R1 � R2 →4R1 � R3 →

�100

10.01

4

10.03

5

���

1,500,00025,500

6,000,000�

�1

0.07�4

10.08

0

10.10

1

1,500,000130,500

0�� x �

0.07x �

�4x �

y �

0.08y �

z0.10z

z

1,500,000130,500

0

�4x � z � 0⇒z � 4x

z �

y �

x �

732 Chapter 8 Matrices and Determinants

86. amount at 9%, amount at 10%,amount at 12%

Solution:

Answer: $100,000 at 9%, $250,000 at 10%, $150,000 at 12%

�100,000, 250,000, 150,000�

x � 2�150,000� � �200,000 ⇒ x � 100,000

y � 3�150,000� � 700,000 ⇒ y � 250,000

�x

y

2z

3z

z

�200,000

700,000

150,000

116R3 →

�1

0

0

0

1

0

�2

3

1

���

�200,000

700,000

150,000�

�R2 � R1 →

7R2 � R3 → �

1

0

0

0

1

0

�2

3

16

���

�200,000

700,000

2,400,000�

100R2 →2R3 →

�1

0

0

1

1

�7

1

3

�5

���

500,000

700,000

�2,500,000�

�0.09R1 � R2 →�2.5R1 � R3 →

�1

0

0

1

0.01

�3.5

1

0.03

�2.5

���

500,000

7,000

�1,250,000�

�1

0.09

2.5

1

0.10

�1

1

0.12

0

���

500,000

52,000

0�

2.5x � y � 0

0.09x � 0.10y � 0.12z � 52,000

x � y z � 500,000

z �y �x � 87.

Equation of parabola: y � x2 � 2x � 5

a � 2 � 5 � 8 ⇒ a � 1

b �32�5� �

192 ⇒ b � 2

c � 5

�a � b

b

c32c

c

8192

5

�12R2 →

�3R2 � R3 → �1

0

0

1

1

0

132

1

���

8192

5�

�4R1 � R2 →

�9R1 � R3 → �1

0

0

1

�2

�6

1

�3

�8

���

8

�19

�52

�1

4

9

1

2

3

1

1

1

���

8

13

20�

� a �

4a �

9a �

b �

2b �

3b �

ccc

81320

y � ax2 � bx � c

88.

6R2 � R3 → �

100

110

132

1

���

9148�

�12R2 → �

100

11

�6

132

�8

���

914

�76�

�4R1 � R2 →�9R1 � R3 →

�1

0

0

1

�2

�6

1

�3

�8

���

9

�28

�76�

�1

4

9

1

2

3

1

1

1

���

9

8

5�

f �3� � 9a � 3b � c � 5

f �2� � 4a � 2b � c � 8

f �1� � a � b � c � 9

f �x� � ax2 � bx � c

Equation of parabola: y � �x2 � 2x � 8

a � �2� � �8� � 9 ⇒ a � �1

b �32�8� � 14 ⇒ b � 2

c � 8

�a � b

b

c32c

c

9

14

8

Section 8.1 Matrices and Systems of Equations 733

89. (a)

Equation of parabola:

(b)

(c) The maximum height is approximately 13 feet and theball strikes the ground at approximately 104 feet.

(d) The maximum occurs at the vertex.

The ball strikes the ground when

By the Quadratic Formula and using the positivevalue for x we have

(e) The values found in part (d) are more accurate, butstill very close to the estimates found in part (c).

x � 103.793 feet.

�0.004x2 � 0.367x � 5 � 0

y � 0.

� 13.418 feet

y � �0.004�45.875�2 � 0.367�45.875� � 5

x � �b

2a�

�0.367

2��0.004�� 45.875

00

120

18

y � �0.004x2 � 0.367x � 5.

a �1

15�1130� �

231125 ⇒ a � � 1

250 � �0.004

�a �115b

b

231125

1130

1225R1 →

�� 130�R2 →

�1

0

115

1

231125

1130�

�4R1 � R2 → �225

015

�30��

4.6�11�

�225

900

15

30��

4.6

7.4�

⇒⇒

225a �

900a �

15b30b

4.67.4

�225a �

900a �

15b �

30b �

ccc

5 9.6

12.4

y � ax2 � bx � c

�0, 5.0�, �15, 9.6�, �30, 12.4� 90. (a)

Equation of parabola:

(b)

(c) For 2003,

When compared to the actual value of 6.3, this is notvery accurate.

(d) For 2008,

The model estimates that in 2008, 24.11 million peo-ple will participate in snowboarding. This indicatesthat the number of participants will almost triple in 5years which is probably not a reasonable estimate.

y � 0.1875�182� � 2.75�18� � 12.86 � 24.11

t � 18.

y � 0.1875�132� � 2.75�13� � 12.86 � 8.8

t � 13.

70

18

28

y � 0.1875t2 � 2.75t � 12.86

a �17��2.75� �

149�12.86� �

235 ⇒ a � 0.1875

b �1663�12.86� �

3160 ⇒ b � �2.75

c �4930 � 63

8 �1029

80 � 12.86

863c �4930

b �1663c �

3160

a �17b �

149c �

235

447 R2 � R3 →�

1

0

0

17

1

0

14916638

63

23531604930�

� 718 R2 →�

1

0

0

17

1

�447

1491663

�7249

2353160

�11370�

�81R1 � R2 →�121R1 � R3 → �

1

0

0

17

�187

�447

149

�3249

�7249

235

�9370

�11370�

149R1 →� 1

81121

17

911

149

11

235

3.35.3

� �

4981

121

79

11

111

2.83.35.3�

�49a

81a

121a

7b

9b

11b

� c

� c

� c

� 2.8

� 3.3

� 5.3

f �11� � 121a � 11b � c � 5.3

f �9� � 81a � 9b � c � 3.3

f �7� � 49a � 7b � c � 2.8

f �x� � at 2 � bt � c

734 Chapter 8 Matrices and Determinants

91. (a)

Let then

Solution:

(b)

(c) s � 0, t � �500: x1 � 0, x2 � �500, x3 � 600, x4 � 500, x5 � 1000, x6 � 0, x7 � �500

s � 0, t � 0: x1 � 0, x2 � 0, x3 � 600, x4 � 0, x5 � 500, x6 � 0, x7 � 0

�s, t, 600 � s, s � t, 500 � t, s, t�

x1 � 600 � �600 � s� � s.x3 � 600 � s, x2 � 500 � �500 � t� � t,x4 � �500 � s � �500 � t� � s � t,

x5 � 500 � t, x7 � t and x6 � s,

x5 � x7 � 500

x4 � x7 � x6 ⇒ x4 � x6 � x7 � 0

x3 � x6 � 600

x2 � x5 � 500

x1 � x2 � x4 ⇒ x1 � x2 � x4 � 0

x1 � x3 � 600

�x1 � x3

x2 � x5

x3 � x6

x4 � x5 � x6

x5 � x7

600

500

600

�500

500

�R2 →�R3 → �

100000

010000

101000

000100

010

�110

001

�100

000010

������

600500600

�500500

0�

�R3 � R2 →�R4 � R3 →

�R4 → �

1

0

0

0

0

0

0

�1

0

0

0

0

1

0

�1

0

0

0

0

0

0

1

0

0

0

�1

0

�1

1

0

0

0

�1

�1

0

0

0

0

0

0

1

0

������

600

�500

�600

�500

500

0�

�R1 � R2 →R2 � R3 →R3 � R4 →R4 � R5 →

�R5 � R6 →

�1

0

0

0

0

0

0

�1

0

0

0

0

1

�1

�1

0

0

0

0

�1

�1

�1

0

0

0

0

1

1

1

0

0

0

0

1

0

0

0

0

0

0

1

0

������

600

�600

�100

500

500

0�

�1

1

0

0

0

0

0

�1

1

0

0

0

1

0

0

1

0

0

0

�1

0

0

1

0

0

0

1

0

0

1

0

0

0

1

�1

0

0

0

0

0

1

1

������

600

0

500

600

0

500�

Section 8.1 Matrices and Systems of Equations 735

92. (a)

Let

Solution:where s and t are real numbers.x1 � 500 � s � t, x2 � �200 � s � t, x3 � s, x4 � 350 � t, x5 � t,

x1 � ��200 � s � t� � 300 ⇒ x1 � 500 � s � t

x2 � s � �350 � t� � 150 ⇒ x2 � �200 � s � t

Let x3 � s.

x4 � t � 350 ⇒ x4 � 350 � t

x5 � t.

�x1

x2

x4

x2

x3 � x4

x5

300

150

350

�R2 →�R3 →

R3 � R4 →

�1

0

0

0

1

1

0

0

0

�1

0

0

0

1

1

0

0

0

1

0

����

300

150

350

0�

R2 � R3 →

�1

0

0

0

1

�1

0

0

0

1

0

0

0

�1

�1

1

0

0

�1

1

����

300

�150

�350

350�

�R1 � R2 → �

1

0

0

0

1

�1

1

0

0

1

�1

0

0

�1

0

1

0

0

�1

1

����

300

�150

�200

350�

�1

1

0

0

1

0

1

0

0

1

�1

0

0

�1

0

1

0

0

�1

1

����

300

150

�200

350�

⇒⇒

x1

x2

x3

x3

x4

x5

150

�200

x1

x1

x2

x4

x2

x3

200

x5

300

150

x3

350

x4

x5

(b) When and

(c) When and

x5 � 350x4 � 0,x3 � 0,x2 � 150,x1 � 150,

150 � �200 � 0 � t ⇒ t � 350.

x2 � �200 � s � t

x3 � 0,x2 � 150

x5 � 350x4 � 0,x3 � 50,x2 � 200,x1 � 100,

200 � �200 � 50 � t ⇒ t � 350.

x2 � �200 � s � t

x3 � 50,x2 � 200

93. False. It is a matrix.2 � 4

94. False. The rows are in the wrong order. To change thismatrix to reduced row-echelon form, interchange Row 1and Row 4, and interchange Row 2 and Row 3.

95. False. Gaussian elimination reduces a matrix until a row-echelon form is obtained and Gauss-Jordan eliminationreduces a matrix until a reduced row-echelon form isobtained.

96.

One possible system is:

or

(Note that the coefficients of x, y, and z have been chosen so that the a-terms cancel.)

�x

x

2x

y

2y

y

7z

11z

10z

�1

0

�3

��4a � 1�2��4a � 1�

��4a � 1�

7a

11a

10a

�1

0

�3�

x

x

2x

y

2y

y

7z

11z

10z

��3a � 2���3a � 2�

2��3a � 2�

x � �3a � 2

y � �4a � 1

z � a

736 Chapter 8 Matrices and Determinants

97. (a) In the row-echelon form of an augmented matrix that corresponds to an inconsistent system of linearequations, there exists a row consisting of all zerosexcept for the entry in the last column.

(b) In the row-echelon form of an augmented matrix that corre-sponds to a system with an infinite number of solutions, thereare fewer rows with nonzero entries than there are variablesand no row has the first non-zero value in the last column.

98. 1. Interchange two rows.

2. Multiply a row by a nonzero constant.

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

100. A matrix in row-echelon form is in reduced row-echelon form if every column that has a leading 1 haszeros in every position above and below its leading 1.

99. They are the same.

101.

Vertical asymptote:

Horizontal asymptote:

Intercept: �2, 0�

y � �2

x � 3

−6−8 2 4 6 8

−4

−6

−8

2

4

6

8

y

x

f �x� �2x2 � 4x3x � x2 �

2x � 43 � x

, x � 0

x 0 1 2 3 4 5

undef. 0 undef. �3�4�1�1.5�1.6f �x�

�1�2

102.

The graph has a vertical asymptote at and a discontinuity at

Since the degrees of the numerator and the denominator are the same, there is a horizontal asymptote at y � 1.

x � 1.x � �1

1

2

3

4

−2−3−4−5 2 3

1

−1

y

x

f �x� �x2 � 2x � 1

x2 � 1�

�x � 1��x � 1��x � 1��x � 1� �

x � 1x � 1

103.

Horizontal asymptote:

Intercept:

−2 −1 1 2 3 4−1

1

2

3

4

5

x

y

�0, 12�y � 0

f�x� � 2x�1

x �1 0 1 2 3

1 2 412

14f �x�

104.

x

2

4

6

8

10

2−2 4 6 8 10−2

y

g�x� � 3�x�2

x 0 1 2 3 4

y 27 9 3 1 19

13

�1

Section 8.2 Operations with Matrices 737

105.

Vertical asymptote:

Intercept: �2, 0�

x � 1

h�x� � ln�x � 1�

x−2 −1 2 3 4 5 6

1

2

3

4

−1

−2

−3

−4

1

y

x 1.5 2 3 4 5

�0.693 0 0.693 1.099 1.386h�x�

106. f �x� � 3 � ln x ⇒ y � 3 � ln x ⇒ e y�3 � x

2 4 6 8 10

−2

2

4

6

8

x

y

x 1

y 0 1 2 3 4

2.720.370.140.05

Section 8.2 Operations with Matrices

■ if and only if they have the same order and

■ You should be able to perform the operations of matrix addition, scalar multiplication, and matrix multiplication.

■ Some properties of matrix addition and scalar multiplication are:

(a)

(b)

(c)

(d)

(e)

(f)

■ You should remember that in general.

■ Some properties of matrix multiplication are:

(a)

(b)

(c)

(d)

■ You should know that the identity matrix of order n, is an matrix consisting of ones on its main diagonal and zeros elsewhere. If A is an matrix, then AIn � InA � A.n � n

n � nIn,

c�AB� � �cA�B � A�cB��A � B�C � AC � BC

A�B � C� � AB � AC

A�BC� � �AB�C

AB � BA

�c � d�A � cA � dA

c�A � B� � cA � cB

1A � A

�cd�A � c�dA�A � �B � C� � �A � B� � C

A � B � B � A

aij � bij.A � B

Vocabulary Check

1. equal 2. scalars 3. zero;

4. identity 5. (a) (iii) (b) (iv) (c) (i) 6. (a) (ii) (b) (iv) (c) (i) (d) (iii)(d) (v) (e) (ii)

O

738 Chapter 8 Matrices and Determinants

1. x � �4, y � 22

3.

x � 2, y � 3

2x � 1 � 5, 3x � 6, 3y � 5 � 4

2. x � 13, y � 12

4.

y � 9 x � �4

y � 2 � 11 2x � �8

�4 � x y � 9

x � 2 � 2x � 6 2y � 18

5. (a)

(b)

(c)

(d) 3A � 2B � �3

6

�3

�3� �2 � 2

�1

�1

8� � �3

6

�3

�3� � ��4

2

2

�16� � ��1

8

�1

�19�

3A � 3�1

2

�1

�1� � �3�1�3�2�

3��1�3��1�� � �3

6

�3

�3�

A � B � �1

2

�1

�1� � � 2

�1

�1

8� � �1 � 2

2 � 1

�1 � 1

�1 � 8� � ��1

3

0

�9�

A � B � �1

2

�1

�1� � � 2

�1

�1

8� � �1 � 2

2 � 1

�1 � 1

�1 � 8� � �3

1

�2

7�

6. (a)

(b)

(c)

(d) 3A � 2B � �3

6

6

3� � 2��3

4

�2

2� � �3 � 6

6 � 8

6 � 4

3 � 4� � � 9

�2

10

�1�

3A � 3�1

2

2

1� � �3�1�3�2�

3�2�3�1�� � �3

6

6

3�

A � B � �1

2

2

1� � ��3

4

�2

2� � �1 � 3

2 � 4

2 � 2

1 � 2� � � 4

�2

4

�1�

A � B � �1

2

2

1� � ��3

4

�2

2� � �1 � 3

2 � 4

2 � 2

1 � 2� � ��2

6

0

3�

7.

(a) (b) (c)

(d) 3A � 2B � �18

6

�9

�3

12

15� � �

2

�2

2

8

10

20� � �

16

8

�11

�11

2

�5�

3A � �18

6

�9

�3

12

15�A � B � �

5

3

�4

�5

�1

�5�A � B � �

7

1

�2

3

9

15�

A � �6

2

�3

�1

4

5�, B � �

1

�1

1

4

5

10�

8. (a)

(b)

(c)

(d) � �2

3

9

�5

�5

16���4

6

6

�2

�8

4�3A � 2B � � 6

�3

3

�3

3

12� � 2� 2

�3

�3

1

4

�2� � � 6

�3

3

�3

3

12� �

3A � 3� 2

�1

1

�1

1

4� � � 3�2�3��1�

3�1�3��1�

3�1�3�4�� � � 6

�3

3

�3

3

12�

� �0

2

4

�2

�3

6�� 2 � 2

�1 � ��3�1 � ��3�

�1 � 1

1 � 4

4 � ��2��A � B � � 2

�1

1

�1

1

4� � � 2

�3

�3

1

4

�2� �

� � 4

�4

�2

0

5

2�A � B � � 2

�1

1

�1

1

4� � � 2

�3

�3

1

4

�2� � � 2 � 2

�1 � 3

1 � 3

�1 � 1

1 � 4

4 � 2�

Section 8.2 Operations with Matrices 739

9.

(a)

(b)

(c)

(d) 3A � 2B � �6

3

6

3

�3

�6

0

0

3

�3� � � 2

�6

2

8

�2

18

2

�12

0

�14� � �4

9

4

�5

�1

�24

�2

12

3

11�

3A � �6

3

6

3

�3

�6

0

0

3

�3�

A � B � �1

4

1

�3

0

�11

�1

6

1

6�

A � B � � 3

�2

3

5

�2

7

1

�6

1

�8�

A � �2

1

2

1

�1

�2

0

0

1

�1�, B � � 1

�3

1

4

�1

9

1

�6

0

�7�

10. (a)

(b)

(c)

(d)

� �35

�5�6

�12

22

3020

�5

�220

�1�10

4� � �

�39

150

�12

12�61224

�3

06

�3�18

0� � �

6�4

�20�6

0

�108

18�4�2

�214284�

3A � 2B � ��3

9150

�12

12�61224

�3

06

�3�18

0� � 2 �

�32

1030

5�4�9

21

1�7�1�4�2

� 3A � 3 �

�1350

�4

4�2

48

�1

02

�1�6

0� � �

�39

150

�12

12�61224

�3

06

�3�18

0�

� ��1 � 3

3 � 25 � 100 � 3

�4 � 0

4 � 5�2 � 4

4 � 98 � 2

�1 � 1

0 � 12 � 7

�1 � 1�6 � 4

0 � 2� � �

21

�5�3�4

�12

136

�2

�190

�22�

A � B � ��1

350

�4

4�2

48

�1

02

�1�6

0� � �

�32

1030

5�4�9

21

1�7�1�4�2

� � �

�1 � 33 � 2

5 � 100 � 3

�4 � 0

4 � 5�2 � 4

4 � 98 � 2

�1 � 1

0 � 12 � 7

�1 � 1�6 � 4

0 � 2� � �

�45

153

�4

9�6�5100

1�5�2

�10�2

� A � B � �

�1350

�4

4�2

48

�1

02

�1�6

0� � �

�32

1030

5�4�9

21

1�7�1�4�2

740 Chapter 8 Matrices and Determinants

11.

(a) is not possible. A and B do not have the same order.

(b) is not possible. A and B do not have the same order.

(c)

(d) is not possible. A and B do not have the same order.3A � 2B

3A � � 18�3

0�12

90�

A � B

A � B

A � � 6�1

0�4

30�, B � �8

4�1�3�

12. (a) is not possible. A and B do not have the same order.

(b) is not possible. A and B do not have the same order.

(c)

(d) is not possible. A and B do not have the same order.3A � 2B

3A � 3�32

�1� � �96

�3�A � B

A � B

13. � ��5 � 7 � ��10�3 � ��2� � 14

0 � 1 � ��8��6 � ��1� � 6� � ��8

15

�7

�1���53

0�6� � � 7

�21

�1� � ��1014

�86�

14. � � 6 � 0 � ��11��1 � ��3� � 2

8 � 5 � ��7�0 � ��1� � ��1�� � ��5

�26

�2�� 6�1

80� � � 0

�35

�1� � ��112

�7�1�

15. � 4��6�3

�18

33� � ��24

�12�432

1212�4���4

002

13� � �2

31

�6�2

0��

17. � � 10�59

89�� �3��6

1503� � � 8

14�8

�18� � � 18�45

0�9� � � 8

14�8

�18��3��07

�32� � ��6

831�� � 2�4

7�4�9�

16.

� �192 2 �7 9

2� � 1

2�19 4 �14 9�

12��5 �2 4 0� � �14 6 �18 9�� �12�5 � 14 �2 � 6 4 � ��18� 0 � 9�

18.

� � �4 �13

2 � ��1��9 � 1

�11 �23

1 �12

�3 � 2� � �

�113

1

�8

�31332

�1�

� ��4

2�9

�111

�3� � �13

�11

2312

2� � �

�42

�9

�111

�3� �16 �

2�6

6

43

12�

�1�4

�2

9

11

�1

3� �

16��

�5

3

0

�1

4

13� � �

7

�9

6

5

�1

�1�� � �

�4

2

�9

�11

1

�3� �

16 �

�5 � 7

3 � ��9�0 � 6

�1 � 5

4 � ��1�13 � ��1��

19. 37� 2

�15

�4� � 6��32

02� ��17.143

11.5712.143

10.286�

Section 8.2 Operations with Matrices 741

20. 55�� 14�22

�1119� � ��22

13206�� � ��440

�495495

1375�

21. � ��1.581

�4.252

9.713

�3.739

�13.249

�0.362���

3.211

�1.004

0.055

6.829

4.914

�3.889� � �

�1.630

5.256

�9.768

�3.090

8.335

4.251�

22. �12��61

�2

20�9

5� � �14

�87

�15�6

0� � ��31

1624

�1910

�10�� � �132

�108�348

1686060�

23. X � 3��2

1

3

�1

0

�4� � 2 �

0

2

�4

3

0

�1� � �

�6

3

9

�3

0

�12� � �

0

4

�8

6

0

�2� � �

�6

�1

17

�9

0

�10�

24.

X � A �12B � �

�2

1

3

�1

0

�4� �

12�

0

2

�4

3

0

�1� � �

�2

1

3

�1

0

�4� � �

0

1

�2

32

0� 1

2� � �

�2

0

5

� 52

0� 7

2�

2X � 2A � B

25. � �3

�12

�132

3

0112

�� �0

1

�2

32

0

�12

�� �3

�32

�92

32

0

6�X � �

32A �

12B � �

32�

�213

�10

�4� �12�

02

�4

30

�1�26.

X � �A � 2B � �1��2

1

3

�1

0

�4� � 2�

0

2

�4

3

0

�1� � �

2

�1

�3

1

0

4� � �

0

�4

8

�6

0

2� � �

2

�5

5

�5

0

6�

2A � 4B � �2X

27. A is and B is is not possible.AB3 � 3.3 � 2 28. A is is is not possible.2 � 2. AB2 � 4, B

29. A is B is

�048

�10

�1

027��

2�3

1

146� � �

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

�8��2� � ��1���3� � �7��1�

�0��1� � ��1��4� � �0��6��4��1� � �0��4� � �2��6�

�8��1� � ��1��4� � �7��6�� � �3

1026

�41646�

3 � 2 ⇒ AB is 3 � 2.3 � 3,

30. A is

AB � ��1

4

0

3

�5

2� �1

0

2

7� � ��1

4

0

19

�27

14�

3 � 2, B is 2 � 2 ⇒ AB is 3 � 2.

742 Chapter 8 Matrices and Determinants

31. A is B is

� �300

0�4

0

00

�10�

�100

040

00

�2��300

0�1

0

005� � �

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

�0��3� � �0��0� � ��2��0�

�1��0� � �0���1� � �0��0��0��0� � �4���1� � �0��0�

�0��0� � �0���1� � ��2��0�

�1��0� � �0��0� � �0��5��0��0� � �4��0� � �0��5�

�0��0� � �0��0� � ��2��5��3 � 3 ⇒ AB is 3 � 3.3 � 3,

32.

AB � �5

0

0

0

�8

0

0

0

7� �

15

0

0

0

�18

0

0

012� � �

1

0

0

0

1

0

0

072�

A is 3 � 3, B is 3 � 3 ⇒ AB is 3 � 3.

33. A is

� �000

000

000�

��0��6� � �0��8� � �5��0�

�0��6� � �0��8� � ��3��0��0��6� � �0��8� � �4��0�

�0���11� � �0��16� � �5��0��0���11� � �0��16� � ��3��0�

�0���11� � �0��16� � �4��0�

�0��4� � �0��4� � �5��0��0��4� � �0��4� � ��3��0�

�0��4� � �0��4� � �4��0���000

000

5�3

4��680

�11160

440� �

3 � 3, B is 3 � 3 ⇒ AB is 3 � 3.

34.

�10

12� �6 �2 1 6� � �60

72

�20

�24

10

12

60

72�A is 2 � 1, B is 1 � 4 ⇒ AB is 2 � 4. 35. �

5

�2

10

6

5

�5

�3

1

5� �

1

8

4

�1

1

�2

2

4

9

� � �41

42

�10

7

5

�25

7

25

45�

36. �11

14

6

�12

10

�2

4

12

9� �

12

�5

15

10

12

16� � �

252

298

217

30

452

180�

37. ��3

�12

5

8

15

�1

�6

9

1

8

6

5� �

3

24

16

8

1

15

10

�4

6

14

21

10� � �

151

516

47

25

279

�20

48

387

87�

38. A is is is not possible.4 � 2. AB3 � 3, B 39. A is and B is is not possible.2 � 4 ⇒ AB2 � 4

40. �15

�4

�8

�18

12

22� ��7

8

22

16

1

24� � ��249

124

232

42

104

176

�417

284

520�

41. (a)

(b)

(c) A2 � �14

22��

14

22� � ��1��1� � �2��4�

�4��1� � �2��4��1��2� � �2��2��4��2� � �2��2�� � � 9

12

6

12�

BA � � 2�1

�18��

14

22� � ��2��1� � ��1��4�

��1��1� � �8��4��2��2� � ��1��2���1��2� � �8��2�� � ��2

31

2

14�

AB � �14

22��

2�1

�18� � ��1��2� � �2���1�

�4��2� � �2���1��1���1� � �2��8��4���1� � �2��8�� � �0

6

15

12�

Section 8.2 Operations with Matrices 743

42. (a)

(b)

(c) A2 � �2

1

�1

4� �2

1

�1

4� � �2�2� � ��1��1�1�2� � 4�1�

2��1� � ��1�41��1� � 4�4�� � �3

6

�6

15�

BA � �0

3

0

�3� �2

1

�1

4� � � 0�2� � �0�13�2� � ��3��1�

0��1� � �0��4�3��1� � ��3�4� � �0

3

0

�15�

AB � �2

1

�1

4��0

3

0

�3� � �2�0� � ��1�31�0� � 4�3�

2�0� � ��1���3�1�0� � 4��3�� � ��3

12

3

�12�

43. (a)

(b)

(c) A2 � �31

�13��

31

�13� � ��3��3� � ��1��1�

�1��3� � �3��1��3���1� � ��1��3�

�1���1� � �3��3�� � �8

6

�6

8�

BA � �13

�31��

31

�13� � ��1��3� � ��3��1�

�3��3� � �1��1��1���1� � ��3��3�

�3���1� � �1��3�� � � 0

10

�10

0�

AB � �31

�13��

13

�31� � ��3��1� � ��1��3�

�1��1� � �3��3��3���3� � ��1��1�

�1���3� � �3��1�� � � 0

10

�10

0�

44. (a)

(b)

(c) A2 � �1

1

�1

1� �1

1

�1

1� � �1�1� � ��1��1�1�1� � �1��1�

1��1� � ��1��1�1��1� � 1�1�� � �0

2

�2

0�

BA � � 1

�3

3

1� �1

1

�1

1� � � 1�1� � �3�1�3�1� � �1��1�

1��1� � 3�1��3��1� � 1�1�� � � 4

�2

2

4�

AB � �1

1

�1

1��1

�3

3

1� � �1�1� � ��1���3�1�1� � 1��3�

1�3� � ��1��1�1�3� � 1�1�� � � 4

�2

2

4�

45. (a)

(b)

(c) is not possible.A2

BA � �1 1 2��78

�1� � ��1��7� � �1��8� � �2���1�� � �13�

AB � �78

�1��1 1 2� � �7�1�8�1�

�1�1�

7�1�8�1�

�1�1�

7�2�8�2�

�1�2�� � �7

8

�1

7

8

�1

14

16

�2�

46. (a)

(b)

(c) The number of columns of A does not equal the number of rows of A; the multiplication is not possible.

BA � �2

3

0� �3 2 1� � �

2�3�3�3�0�3�

2�2�3�2�0�2�

2�1�3�1�0�1�� � �

6

9

0

4

6

0

2

3

0�

AB � �3 2 1� �2

3

0� � �3�2� � 2�3� � 1�0�� � �12�

47. �30

1�2��

1�2

02��

12

04� � �1

42

�4��12

04� � � 5

�48

�16�

48.

� � 27�6

�6�27�

� �3��92

29�

3��61

5�2

�10��

0�1

4

3�3

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

6�3� � 5��3� � ��1��1�1�3� � ��2���3� � �0��1���

744 Chapter 8 Matrices and Determinants

49. �04

21

�22���

40

�1

0�1

2� � ��2�3

0

35

�3�� � �04

21

�22��

2�3�1

34

�1� � ��43

1014�

50.

� �3�4�

��1��4�5�4�7�4�

3�2���1��2�

5�2�7�2�

� � �12

�42028

6�21014

�3

�157���5 �6� � �7 �1� � ��8 9�� � �

3�1

57��4 2�

51. (a)

(b)

X � �48�

2R1 � R2 → �10

01

��

48�

�R2 � R1 →

�1

�2

0

1��

4

0�

��1

�2

1

1��

4

0�

��1

�2

1

1��x1

x2� � �4

0�

53. (a)

(b)

X � ��76�

32R2 � R1 → �1

001

��

�76�

12R1 →

�18R2 → �1

0

32

1��

26�

3R1 � R2 → ��20

�3�8

��

�4�48�

��26

�31

��

�4�36�

��26

�31��

x1

x2� � � �4

�36�

52. (a)

(b)

X � ��2

3�

�4R2 � R1 →

�15R2 →

�10

01

��

�23�

15R2 →�1

041

��

103�

�2R1 � R2 → �1

04

�5��

10�15�

R2

R1

�1

2

4

3��

10

5�

�2

1

3

4��x1

x2� � � 5

10�

54. (a)

(b)

X � ��23

�353�

3R2 � R1 → �1

001

��

�23�

353�

�13R2 →

�10

�31

��

12�

353�

4R1 � R2 → �1

0�3�3

��

1235�

R1

R2

� 1

�4

�3

9��

12

�13�

��4

1

9

�3��x1

x2� � ��13

12�

Section 8.2 Operations with Matrices 745

55. (a)

(b)

X � �1

�12�

�7R3 � R1 →�2R3 � R2 → �

100

010

001

���

1�1

2�

2R2 � R1 →

R2 � R3 → �

100

010

721

���

1532�

R1 � R2 →�2R2 � R3 →

�100

�21

�1

32

�1

���

93

�1�

�1

�1

2

�2

3

�5

3

�1

5

���

9

�6

17�

A � �1

�1

2

�2

3

�5

3

�1

5��

x1

x2

x3� � �

9

�6

17� 56. (a)

(b)

X � �03

�2�

2R3 � R1 →

R3 � R2 → �100

010

001

���

03

�2�

�R2 � R1 →

�R3 → �100

010

�2�1

1

���

45

�2�

�R2 � R3 → �

100

110

�3�1�1

���

952�

13R2 →

�12R3 → �

100

111

�3�1�2

���

957�

R1 � R2 →�R1 � R3 → �

100

13

�2

�3�3

4

���

915

�14�

�1

�11

12

�1

�301

���

96

�5��

1�1

1

12

�1

�301� �

x1

x2

x3� � �

96

�5�

57. (a)

(b)

X � ��1

3�2�

�2R3 � R1 →

�100

010

001

���

�13

�2�

15R3 →

�100

010

201

���

�53

�2�

5R2 � R1 →

2R2 � R3 → �

100

010

205

�53

�10�

�1

12R2 → �100

�51

�2

205

�203

�16�

�R3 � R2 → �100

�5�12�2

205

�20�36�16�

3R1 � R2 → �100

�5�14�2

255

�20�52�16�

�1

�30

�51

�2

2�1

5

���

�208

�16��

1�3

0

�51

�2

2�1

5��x1

x2

x3� � �

�208

�16� 58. (a)

(b)

X � �4

�5

2�

�3R3 � R1 →

R3 � R2 → �100

010

001

���

4�5

2�

R2 � R1 →

�R3 → �

1

0

0

0

1

0

3

�1

1

���

10

�7

2�

6R2 � R3 → �

1

0

0

�1

1

0

4

�1

�1

���

17

�7

�2�

14R2 → �

1

0

0

�1

1

�6

4

�1

5

���

17

�7

40�

�R1 � R2 → �100

�14

�6

4�4

5

���

17�28

40�

�1

1

0

�1

3

�6

4

0

5

���

17

�11

40�

�1

1

0

�1

3

�6

4

0

5��

x1

x2

x3� � �

17

�11

40�

746 Chapter 8 Matrices and Determinants

59. 1.2�7035

50100

2570� � �84

4260

1203084� 60. 1.10�100

40

90

20

70

60

30

60� � �110

44

99

22

77

66

33

66�

61. (a) Farmer’s Fruit FruitMarket Stand Farm

Each entry represents the number of bushels of each type of crop that are shipped to each outlet.

(b)

Each entry represents the profit per bushel for each type of crop.

(c)

The entries in the matrix represent the profits for both crops at each of the three outlets.

� �$1037.50 $1400.00 $1012.50�

BA � �3.50 6.00��125100

100175

75125�

B � �3.50 6.00�

75125�

Apples Peaches

100175

A � �125100

62.

The entries represent the costs of the three models of the product at the two warehouses.

BA � �$39.50 $44.50 $56.50��5,000

6,000

8,000

4,000

10,000

5,000� � �$916,500 $885,500�

63.

The entries represent the wholesale and retail inventory values of the inventories at the three outlets.

ST � �304

222

231

343

032��

840

1200

1450

2650

3050

1100

1350

1650

3000

3200� � �

$15,770

$26,500

$21,260

$18,300

$29,250

$24,150�

64.

The matrix gives the proportion of the voting population that changed parties or remained loyal to their party from the first election to the third.

P2

P2 � �0.6

0.2

0.2

0.1

0.7

0.2

0.1

0.1

0.8� �

0.6

0.2

0.2

0.1

0.7

0.2

0.1

0.1

0.8� � �

0.40

0.28

0.32

0.15

0.53

0.32

0.15

0.17

0.68�

65.

—CONTINUED—

P6 � �0.213

0.311

0.477

0.197

0.326

0.477

0.197

0.280

0.523�

P5 � P4P � �0.250

0.315

0.435

0.188

0.377

0.435

0.188

0.248

0.565� �

0.6

0.2

0.2

0.1

0.7

0.2

0.1

0.1

0.8� � �

0.225

0.314

0.461

0.194

0.345

0.461

0.194

0.267

0.539�

P4 � P3P � �0.300

0.308

0.392

0.175

0.433

0.392

0.175

0.217

0.608� �

0.6

0.2

0.2

0.1

0.7

0.2

0.1

0.1

0.8� � �

0.250

0.315

0.435

0.188

0.377

0.435

0.188

0.248

0.565�

P3 � P2P � �0.40

0.28

0.32

0.15

0.53

0.32

0.15

0.17

0.68� �

0.6

0.2

0.2

0.1

0.7

0.2

0.1

0.1

0.8� � �

0.300

0.308

0.392

0.175

0.433

0.392

0.175

0.217

0.608�

Section 8.2 Operations with Matrices 747

65. —CONTINUED—

As P is raised to higher and higher powers, the resulting matrices appear to be approaching the matrix

�0.2

0.3

0.5

0.2

0.3

0.5

0.2

0.3

0.5�.

P8 � �0.203

0.305

0.492

0.199

0.309

0.492

0.199

0.292

0.508�

P7 � �0.206

0.308

0.486

0.198

0.316

0.486

0.198

0.288

0.514�

66.

This represents the labor cost for each boat size at each plant.

ST � �1

1.6

2.5

0.5

1.0

2.0

0.2

0.2

1.4� �

12

9

8

10

8

7� � �

$18.10

$29.80

$59.20

$15.40

$25.40

$50.80�

Sales Profit

67. (a)

The entries in Column 1 represent the total sales of the three kinds of milk for Friday, Saturday, and Sunday. The entries in Column 2 represent each days’ total profit.

(b) Total profit for the weekend: 115 � 161 � 188 � $464

� �447624.50731.20

115161188�

Friday Saturday Sunday

AB � �406076

648296

527684� �

2.652.853.05

0.650.700.85�

69. (a)

(b)

The first entry represents the total calories burned by the 120-pound person and the second entry represents the total calories burned by the 150-pound person.

BA � �2 0.5 3� �10912764

13615979� �

120-poundperson[473.5

150-poundperson588.5] Calories burned

B �

Bicycled

[2

Jogged

0.5

Walked

3]

20-minute time periods

Sales ($) Profit

68. (a)

The first column of AB gives the amount of sales for each octane. The second column gives the profit made by each octane.

(b) The store’s profit for the weekend is $616 � $394.40 � $858.40 � $1868.80.

� �354122974929

616394.4858.4�

87 89 93

AB � �580 560 860

840420

1020

320160540� �

1.952.052.15

0.320.360.40�

70. (a) Individual Familycosts costs

—CONTINUED—

B � �683.91463.1499.27

1699.481217.451273.08�

Comprehensive planHMO standard plan HMO plus plan

A � �694.32451.8489.48

1725.361187.761248.12�

Comprehensive planHMO standard plan HMO plus plan

748 Chapter 8 Matrices and Determinants

70. —CONTINUED—(b) Change in Change in

individual family costs cost

Employees choosing the comprehensive plan have a decrease in cost while those choosing the other two have an increased cost.

(c) Dividing each entry of matrix A by 12 yields

(d) If the costs increase by 4% next year, then the new cost matrix would be:

Monthly Monthlyindividual family

cost cost

112�A � 0.04A� � �

60.1739.1642.42

149.53102.94108.17�

Comprehensive planHMO standard plan HMO plus plan

A � 0.04A � �722.09469.87509.06

1794.371235.271298.05�

112B � �

56.9938.5941.61

141.62101.45106.09�.1

12A � �57.8637.6540.79

143.7898.98

104.01�,

�10.41

�11.3�9.79

25.88�29.69�24.96�

Comprehensive planHMO standard plan HMO plus plan

� �683.91463.1499.27

1699.481217.451273.08� �A � B � �

694.32451.8489.48

1725.361187.761248.12�

71. True. The sum of two matrices of different orders is undefined.

72. False. For most matrices, AB � BA.

For 73–80, A is of order B is of order C is of order and D is of order 2 � 2.3 � 22 � 3,2 � 3,

73. is not possible. A and C are not of the same order.A � 2C 74. is not possible. B and C are not of the same order.B � 3C

75. AB is not possible. The number of columns of A does not equal the number of rows of B.

76. BC is possible. The resulting order is 2 � 2.

77. is possible. The resulting order is 2 � 2.BC � D 78. is not possible. The order of is but theorder of D is 2 � 2.

3 � 3,CBCB � D

79. is possible. The resulting order is 2 � 3.D�A � 3B� 80. is possible. The resulting order is 2 � 3.�BC � D�A

81.

Thus, AC � BC even though A � B.

BC � �1

1

0

0��2

2

3

3� � �2

2

3

3�

AC � �0

0

1

1��2

2

3

3� � �2

2

3

3� 82.

and neither A nor B is O.AB � O

AB � �34

34��

1�1

�11� � �0

000�

83. The product of two diagonal matrices of the same order is a diagonal matrix whose entries are the products of the corresponding diagonal entries of A and B.

Section 8.2 Operations with Matrices 749

84. (a)

(b)

the identity matrixB2 � �0

i

�i

0� �0

i

�i

0� � ��0��0� � ��i��i��i��0� � �0��i�

�0���i� � ��i��0��i���i� � �0��0� � � �1

0

0

1� � I,

B � �0

i

�i

0�

A4 � A3A � ��i

0

0

�i��i

0

0

i� � ���i��i� � �0��0��0��i� � ��i��0�

�i��0� � �0��i��0��0� � ��i��i�� � �1

0

0

1� and i4 � 1

A3 � A2A � ��1

0

0

�1��i

0

0

i� � ���1��i� � �0��0��0��i� � ��1��0�

��1��0� � �0��i��0��0� � ��1��i�� � ��i

0

0

�i� and i3 � �i

A2 � � i

0

0

i��i

0

0

i� � ��i��i� � �0��0��0��i� � �i��0�

�i��0� � �0��i��0��0� � �i��i�� � ��1

0

0

�1� and i2 � �1

85.

Solutions: 43, �8

x �43 or x � �8

3x � 4 � 0 or x � 8 � 0

�3x � 4��x � 8� � 0

3x2 � 20x � 32 � 0 86.

Solutions: �14, 32

4x � 1 � 0 ⇒ x � �14

2x � 3 � 0 ⇒ x �32

�2x � 3��4x � 1� � 0

8x2 � 10x � 3 � 0

87.

by the Quadratic Formula

Solutions: 0, �5 ± 37

4

��5 ± 37

4

x ��10 ± 102 � 4�4���3�

2�4� ��10 ± 148

8

x � 0 or 4x2 � 10x � 3 � 0

x�4x2 � 10x � 3� � 0

4x3 � 10x2 � 3x � 0 88.

Solutions: 0, �9, 53

3x � 5 � 0 ⇒ x �53

x � 9 � 0 ⇒ x � �9

x � 0

x�x � 9��3x � 5� � 0

x�3x2 � 22x � 45� � 0

3x3 � 22x2 � 45x � 0

89.

or

Solutions: 4, ±15

3i

x � ±�53

� ±15

3i

x2 � �53

x � 4

3x2 � 5 � 0 x � 4 � 0

�x � 4��3x2 � 5� � 0

3x2�x � 4� � 5�x � 4� � 0

3x3 � 12x2 � 5x � 20 � 0 90.

Solutions:52

, ±6

x � ±6

x2 � 6 � 0 ⇒ x2 � 6 ⇒ x � ±6

2x � 5 � 0 ⇒ x �52

�2x � 5��x2 � 6� � 0

x2�2x � 5� � 6�2x � 5� � 0

2x3 � 5x2 � 12x � 30 � 0

91.

Solution: �7, �12�

�x � 4��12� � �9 ⇒ x � 7

�5�Eq.1

Add equations.

�5x5x

20y8y

12yy

�4539�6�

12

Eq.1

Eq.2��x

5x

4y

8y

�9

39

( )7, − 12

2−2−4

2

4

−4

−8

−10

x

y

750 Chapter 8 Matrices and Determinants

92.

Solution: ��1, 3�

8x � 3�3� � �17 ⇒ x � �1

�6�Eq.1

�8�Eq.2

Add equations.

48x�48x

18y56y38y

y

� �102� 216� 114� 3

Equation 1

Equation 2� 8x

�6x

3y

7y

�17

27

x2 3 41−1−3−4

1

2

5

6

7

(−1, 3)

y

93.

Solution: �3, �1�

�3 � 2y � �5 ⇒ y � �1

�2�Eq.2

Add equations.

�x

�6x

�7x

2y

2y

x

�5

�16

�21

3

Equation 1

Equation 2� �x

�3x

2y

y

�5

�8

x6 8−2−4

2

−2

−4

−6

−8

4

y

(3, −1)

94.

Solution: �4, 1�

6x � 13�1� � 11 ⇒ x � 4

�3�Eq.1

��2�Eq.2

Add equations.

18

�18

x

x

39y

10y

�49y

y

33

�82

�49

1

Equation 1

Equation 2� 6x

9x

13y

5y

11

41

(4, 1)

2−2−4

2

4

6

8

4 86−2

−4

x

y

Section 8.3 The Inverse of a Square Matrix

■ You should know that the inverse of an matrix A is the matrix if is exists, such that where I is the identity matrix.

■ You should be able to find the inverse, if it exists, of a square matrix.

(a) Write the matrix that consists of the given matrix A on the left and the identity matrix I on the rightto obtain Note that we separate the matrices A and I by a dotted line. We call this process adjoining thematrices A and I.

(b) If possible, row reduce A to I using elementary row operations on the entire matrix The result will bethe matrix If this is not possible, then A is not invertible.

(c) Check your work by multiplying to see that

■ The inverse of is if

■ You should be able to use inverse matrices to solve systems of linear equations if the coefficient matrix is square and invertible.

ad � cb � 0.A�1 �1

ad � bc�d

�c�b

a�A � �ac

bd�

AA�1 � I � A�1A.

�I � A�1�.�A � I�.

�A � I�.n � nn � 2n

n � nAA�1 � A�1A � I,A�1,n � nn � n

Vocabulary Check

1. square 2. inverse 3. nonsingular; singular 4. A�1B

Section 8.3 The Inverse of a Square Matrix 751

1.

BA � � 3�5

�12��

25

13� � � 6 � 5

�10 � 10

3 � 3

�5 � 6� � �1

0

0

1�

AB � �25

13��

3�5

�12� � � 6 � 5

15 � 15

�2 � 2

�5 � 6� � �1

0

0

1�

2.

BA � �2

1

1

1��1

�1

�1

2� � �2 � 1

1 � 1

�2 � 2

�1 � 2� � �1

0

0

1�

AB � � 1

�1

�1

2��2

1

1

1� � � 2 � 1

�2 � 2

1 � 1

�1 � 2 � � �1

0

0

1�

3.

BA � ��232

1

�12� �1

3

2

4� � ��2 � 332 �

32

�4 � 4

3 � 2� � �1

0

0

1�

AB � �1

3

2

4� ��232

1

�12� � ��2 � 3

�6 � 6

1 � 1

3 � 2� � �1

0

0

1�

4.

BA � �35

� 25

1515� �1

2

�1

3� � �35 �

25

�25 �

25

�35 �

35

25 �

35� � �1

0

0

1�

AB � �1

2

�1

3��35

� 25

1515� � �

35 �

25

65 �

65

15 �

15

25 �

35� � �1

0

0

1�

5.

BA � �123

146

2�3�5��

2�1

0

�17113

11�7�2� � �

2 � 1

4 � 4

6 � 6

�17 � 11 � 6

�34 � 44 � 9

�51 � 66 � 15

11 � 7 � 4

22 � 28 � 6

33 � 42 � 10� � �

1

0

0

0

1

0

0

0

1�

AB � �2

�10

�17113

11�7�2��

123

146

2�3�5� � �

2 � 34 � 33

�1 � 22 � 21

6 � 6

2 � 68 � 66

�1 � 44 � 42

12 � 12

4 � 51 � 55

�2 � 33 � 35

�9 � 10� � �

1

0

0

0

1

0

0

0

1�

6.

� �1

0

0

0

1

0

0

0

1�� �

2 � 1

�1 � 1

1 � 1

�12 � 2 �

32

14 � 2 �

114

�14 � 2 �

74

�52 � 4 �

32

54 � 4 �

114

�54 � 4 �

74� BA � ��

1214

�14

1

�1

1

32

�11474���4

�1

0

1

2

�1

5

4

�1�

� �1

0

0

0

1

0

0

0

1�� �

2 �14 �

54

12 �

12 � 1

�14 �

14

�4 � 1 � 5

�1 � 2 � 4

1 � 1

�6 �114 �

354

�32 �

112 � 7114 �

74� AB � �

�4�1

0

12

�1

54

�1���

1214

�14

1

�1

1

32

�11474�

7.

—CONTINUED—

� ��2 � 3

0

1 � 4 � 3

0

4 � 1 � 5

6 � 5

�2 � 9 � 2 � 5

8 � 9 � 1

�2 � 1 � 3

0

1 � 5 � 2 � 3

�4 � 5 � 1

�2 � 1 � 3

0

1 � 6 � 2 � 3

�4 � 6 � 1� � �

1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1�

AB � �2

3

�1

4

0

0

1

�1

1

0

�2

1

1

1

1

0� �

�1

�4

0

3

2

9

1

�5

�1

�5

�1

3

�1

�6

�1

3�

752 Chapter 8 Matrices and Determinants

7. —CONTINUED—

� ��2 � 6 � 1 � 4

�8 � 27 � 5 � 24

3 � 1 � 4

6 � 15 � 3 � 12

0

�5 � 6

0

0

�1 � 2 � 1

�4 � 10 � 6

2 � 1

3 � 6 � 3

�1 � 2 � 1

�4 � 9 � 5

0

3 � 5 � 3� � �

1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1�

BA � ��1

�4

0

3

2

9

1

�5

�1

�5

�1

3

�1

�6

�1

3� �

2

3

�1

4

0

0

1

�1

1

0

�2

1

1

1

1

0�

8.

� �1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1�

� �6 � 3 � 2

�24 � 14 � 10

10 � 6 � 4

6 � 4 � 2

3 � 1 � 2

�14 � 5 � 10

6 � 2 � 4

4 � 1 � 3

�3 � 9 � 6

12 � 42 � 30

�5 � 18 � 12

�3 � 12 � 9

�2 � 2

10 � 10

�4 � 4

�2 � 3�

BA � ��3

12

�5

�3

�3

14

�6

�4

1

�5

2

1

�2

10

�4

�3��

�2

1

�2

0

0

�1

�1

1

1

�3

0

3

0

0

�2

�1�

� �1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1�

� �6 � 5

�3 � 12 � 15

6 � 12 � 6

12 � 15 � 3

6 � 6

�3 � 14 � 18

6 � 14 � 8

14 � 18 � 4

�2 � 2

1 � 5 � 6

�2 � 5 � 2

�5 � 6 � 1

4 � 4

�2 � 10 � 12

4 � 10 � 6

10 � 12 � 3�

AB � ��2

1

�2

0

0

�1

�1

1

1

�3

0

3

0

0

�2

�1��

�3

12

�5

�3

�3

14

�6

�4

1

�5

2

1

�2

10

�4

�3�

9.

BA �13�

�4

�4

1

�5

�8

2

3

3

0� �

�2

1

0

2

�1

1

3

0

4� �

13 �

8 � 5

8 � 8

�2 � 2

�8 � 5 � 3

�8 � 8 � 3

2 � 2

�12 � 12

�12 � 12

3� � �

1

0

0

0

1

0

0

0

1�

AB �13�

3

0

0

0

3

0

0

0

3� � �

1

0

0

0

1

0

0

0

1�

AB �13�

�2

1

0

2

�1

1

3

0

4� �

�4

�4

1

�5

�8

2

3

3

0� �

13 �

8 � 8 � 3

�4 � 4

�4 � 4

10 � 16 � 6

�5 � 8

�8 � 8

�6 � 6

3 � 3

3�

Section 8.3 The Inverse of a Square Matrix 753

10.

� �1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1�

� 13 �

3 � 1 � 13 � 1 � 2

1 � 13 � 2 � 1

�3 � 1 � 1 � 3�3 � 1 � 2 � 3

�1 � 1�3 � 2 � 1

1 � 2 � 3�1 � 4 � 3

1 � 2�2 � 2

3 � 33 � 3

03�

BA �13 �

�3�3

0�3

1�1

1�2

1211

�3�3

00��

�11

�10

1�1

1�1

0121

�1001�

� �1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1�

� 13 �

3 � 3 � 3

�3 � 3

3 � 3

3 � 3

�1 � 1 � 2

1 � 1 � 1

�1 � 1 � 2

1 � 1 � 2

�1 � 2 � 1

1 � 2 � 1

�1 � 2 � 2

�2 � 1 � 1

3 � 3

�3 � 3

3 � 3

3�

AB �13 �

�11

�10

1�1

1�1

0121

�1001��

�3�3

0�3

1�1

1�2

1211

�3�3

00�

11.

A�1 � �12

0

013�

12R1 →

13R2 → �

1

0

0

1

12

0

013� � �I � A�1�

�A � I� � �2

0

0

3��

1

0

0

1�

13.

A�1 � ��3

�2

2

1�

2R2 � R1 → �1

001

��

�3�2

21� � �I � A�1�

�2R1 � R2 → �10

�21

��

1�2

01�

�A � I� � �1

2

�2

�3��

1

0

0

1�

12.

A�1 � � 7

�3

�2

1�

�2R2 � R1 → �1

0

0

1��

7

�3

�2

1� � �I � A�1�

�3R1 � R2 →

�1

0

2

1��

1

�3

0

1�

�A � I� � �1

3

2

7��

1

0

0

1�

14.

A�1 � ��19

�4

�33

�7�

5R2 � R1 → �1

0

0

1��

�19

�4

�33

�7� � �I � A�1�

�4R1 � R2 →

�1

0

�5

1��

1

�4

2

�7�

2R2 � R1 → �1

4

�5

�19��

1

0

2

1�

�A � I� � ��7

4

33

�19��

1

0

0

1�

754 Chapter 8 Matrices and Determinants

15.

A�1 � �1

2

�1

�1�

2R1 � R2 → �1

0

0

1��

1

2

�1

�1� � �I ... A�1�

�R2 � R1 → � 1

�201

��

10

�11�

�A � I� � ��1

�2

1

1��

1

0

0

1� 16.

A�1 � �0

1

�1

11�

�R2 � R1 → �1

0

0

1��

0

1

�1

11� � �I � A�1�

R1 � R2 →

�1

0

1

1��

1

1

10

11�

10R2 � R1 → � 1

�1

1

0��

1

0

10

1�

�A � I� � � 11

�1

1

0��

1

0

0

1�

17.

The two zeros in the second row imply that the inverse does not exist.

�2R1 � R2 → �20

40

��

1�2

01�

�A � I� � �2

4

4

8��

1

0

0

1�

19. A � � 2

�3

7

�9

1

2� 20. A � ��2

6

0

5

�15

1�

21.

A�1 � �1

�3

3

1

2

�3

�1

�1

2�

2R3 → �

100

010

001

���

1�3

3

12

�3

�1�1

2� � �I � A�1�

�R3 � R1 →�R3 � R2 → �

1

0

0

0

1

0

0

012

���

1

�332

1

2

�32

�1

�1

1�

�R2 � R1 →

�3R2 � R3 →

�1

0

0

0

1

0

121212

52

�3232

�1212

�32

0

0

1�

12R2 → �

100

113

112

2

���

1�

32

�3

012

0

001�

�3R1 � R2 →�3R1 � R3 →

�100

123

112

���

1�3�3

010

001�

�A � I� � �1

3

3

1

5

6

1

4

5

���

1

0

0

0

1

0

0

0

1�

A has no inverse becauseit is not square.

A has no inverse because itis not square.

18.

A�1 �15� 4

�1

�3

2�

�4R2 � R1 → �1

0

0

1

��

45

� 15

� 3525� � �I � A�1�

� 1

5R2 → �1

0

4

1��

0� 1

5

125�

�2R1 � R2 →

�1

0

4

�5��

0

1

1

�2�

R2

R1

�1

2

4

3��

0

1

1

0�

�A � I� � �2

1

3

4��

1

0

0

1�

Section 8.3 The Inverse of a Square Matrix 755

22.

A�1 � ��13

12

�5

6

�5

2

4

�3

1�

4R3 � R1 →�3R3 � R2 → �

1

0

0

0

1

0

0

0

1

���

�13

12

�5

6

�5

2

4

�3

1� � �I � A�1�

�2R2 � R1 →

2R2 � R3 → �

1

0

0

0

1

0

�4

3

1

���

7

�3

�5

�2

1

2

0

0

1�

�3R1 � R2 →R1 � R3 →

�1

0

0

2

1

�2

2

3

�5

���

1

�3

1

0

1

0

0

0

1�

�A � I� � �1

3

�1

2

7

�4

2

9

�7

���

1

0

0

0

1

0

0

0

1�

23.

A�1 � �1

�347

20

014

�14

0

015

� 14R2 → 15R3 → �

10

0

01

0

00

1

���

1�

34720

014

�14

0015

� � �I � A�1�

�54R2 � R3 →

�1

0

0

0

4

0

0

0

5

���

1

�374

0

1

�54

0

0

1�

�3R1 � R2 →

�2R1 � R3 → �1

0

0

0

4

5

0

0

5

1

�3

�2

0

1

0

0

0

1�

�A � I� � �1

3

2

0

4

5

0

0

5

���

1

0

0

0

1

0

0

0

1�

24.

Since the first three entries of row 2 are all zeros, the inverse of A does not exist.

�3R1 � R2 →�2R1 � R3 →

�1

0

0

0

0

5

0

0

5

���

1

�3

�2

0

1

0

0

0

1��A � I� � �

1

3

2

0

0

5

0

0

5

���

1

0

0

0

1

0

0

0

1�

25.

A�1 � ��

18

0

0

0

0

1

0

0

0

014

0

0

0

0

�15

�18R1 →

14R3 →

�15R4 →

�1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1

����

�18

0

0

0

0

1

0

0

0

014

0

0

0

0

�15

� � �I � A�1�

�A � I� � ��8

0

0

0

0

1

0

0

0

0

4

0

0

0

0

�5

����

1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1�

756 Chapter 8 Matrices and Determinants

26.

A�1 �110�

10

0

0

0

�15

5

0

0

�40

10

�5

0

26

�8

1

2�

9R4 � R1 →

�1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1

1

0

0

0

� 3212

0

0

�4

1

� 12

0

135

� 45

11015

� � �I � A�1�

�4R3 � R1 →�4R4 � R2 →

�12R3 →

�1

0

0

0

0

1

0

0

0

0

1

0

�9

0

0

1

���

1

0

0

0

� 3212

0

0

�4

1

� 12

0

45

� 45

11015

�3R2 � R1 →R3 � R2 →

�R4 � R3 → �

1

0

0

0

0

1

0

0

�8

0

�2

0

�9

4

0

1

����

1

0

0

0

� 3212

0

0

0

1

1

0

0

0� 1

515

12R2 →

15R4 →

�1

0

0

0

3

1

0

0

�2

2

�2

0

0

3

1

1

����

1

0

0

0

012

0

0

0

0

1

0

0

0

015

�A � I� � �1

0

0

0

3

2

0

0

�2

4

�2

0

0

6

1

5

����

1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1�

27.

A�1 � ��175

95

14

37

�20

�3

�13

7

1�

A � �1

3

�5

2

7

�7

�1

�10

�15�

29.

A�1 �12�

�3

9

�2

3

�7

2

2

�6

2� � �

�1.5

4.5

�1

1.5

�3.5

1

1

�3

1�

A � �1

3

�2

1

1

0

2

0

3�

28.

A�1 � ��10

2

�13

�4

1

�5

27

�5

35�

A � �10

�5

3

5

1

2

�7

4

�2�

30.

A�1 � �17

�8

�1�8.5

10

01

�1�

A � �3

2

�4

2

2

4

2

2

3�

31.

A�1 � ��12�4�8

�5�2�4

�9�4�6�

A � ��12

1

0

34

0

�1

14

�3212

� 32.

does not exist.A�1

��

56

0

1

13

23

�12

116

2

�52

Section 8.3 The Inverse of a Square Matrix 757

33.

A�1 �511�

0

�22

22

�4

11

�6

2

11

�8� � �

0

�10

10

�1.81

5

�2.72

0.90

5

�3.63�

A � �0.1

�0.3

0.5

0.2

0.2

0.4

0.3

0.2

0.4� 34.

A�1 � �3.75

3.45834.16

0�1

0

�1.25�1.375

�2.5�

A � �0.60.7

1

0�1

0

�0.30.2

�0.9�

35.

does not exist.A�1

A � �1

0

1

0

0

2

0

2

3

0

3

0

0

4

0

4� 36.

A�1 � �27

�16

�17

�7

�10

5

4

2

4

�2

�2

�1

�29

18

20

8�

A � �4

2

0

3

8

5

2

6

�7

�4

1

�5

14

6

�7

10�

37.

A�1 � �1

0

2

0

0

1

0

1

1

0

1

0

0

1

0

2�

A � ��1

0

2

0

0

2

0

�1

1

0

�1

0

0

�1

0

1� 38.

A�1 � ��24

�10

�29

12

7

3

7

�3

1

0

3

�1

�2

�1

�2

1�

A � �1

3

2

�1

�2

�5

�5

4

�1

�2

�2

4

�2

�3

�5

11�

39.

A�1 �1

19�3

�225� � �

319

�219

219519�

ad � bc � �5��3� � ��2��2� � 19

A � �5

2

�2

3�

A � �a

c

b

d�, A�1 �1

ad � bc � d

�c

�b

a� 40.

A�1 �1

61��5

8

�12

7� � ��5

61

861

�1261

761�

ad � bc � 7��5� � 12��8� � �35 � 96 � 61

A � � 7

�8

12

�5�

41.

Since does not exist.A�1ad � bc � 0,

ad � bc � ��4��3� � ��2���6� � 0

A � ��42

�63� 42.

A�1 �19�

�2

�5

�3

�12� � ��29

�59

�13

�43�

ad � bc � ��12���2� � 3�5� � 24 � 15 � 9

A � ��12

5

3

�2�

43.

A�1 �1

59�20 �45

�15

3472� �

2059�

45

�15

3472� � �

1659

�459

15597059�

ad � bc � 72

45 � �

34

15 �

2810

�3

20�

5920

A � �7215

�3445� 44.

A�1 � �36143�

89

�53

�94

�14� � ��

3214360

143

811439

143�

ad � bc � �14

89 � 9

453 � �

14336

A � ��14

53

94

89�

758 Chapter 8 Matrices and Determinants

45.

Solution: �5, 0�

�x

y� � ��3

�2

2

1� � 5

10� � �5

0� 46.

Solution: �6, 3�

�x

y� � ��3

�2

2

1� �0

3� � �6

3�

47.

Solution: ��8, �6�

�x

y� � ��3

�2

2

1� �4

2� � ��8

�6� 48.

Solution: ��7, �4�

�x

y� � ��3

�2

2

1��1

�2� � ��7

�4�

49.

Solution: �3, 8, �11�

�x

y

z� � �

1

�3

3

1

2

�3

�1

�1

2� �

0

5

2� � �

3

8

�11�

51.

Solution: �2, 1, 0, 0�

�x1

x2

x3

x4

� � ��24

�10

�29

12

7

3

7

�3

1

0

3

�1

�2

�1

�2

1� �

0

1

�1

2� � �

2

1

0

0�

53.

Solution: �2, �2�

�x

y� � �1

11 � 3

�5

�4

3� ��2

4� � �1

11 ��22

22� � � 2

�2�

A�1 �1

9 � 20 � 3

�5

�4

3�

A � �3

5

4

3�

50.

Solution: �1, 7, �9�

�x

y

z� � �

1

�3

3

1

2

�3

�1

�1

2��

�1

2

0� � �

1

7

�9�

52.

Solution: ��32, �13, �37, 15�

�x

y

z

w� � �

�24

�10

�29

12

7

3

7

�3

1

0

3

�1

�2

�1

�2

1��

1

�2

0

�3� � �

�32

�13

�37

15�

54.

Solution: �12, 13�

�1

72�36

24� � �12

13��x

y� �1

72�24

�30

�12

18��13

23�

A�1 �1

432 � 360 � 24

�30

�12

18�

A � �18

30

12

24�

55.

This implies that there is no unique solution; that is,either the system is inconsistent or there are infinitelymany solutions.

Find the reduced row-echelon form of the matrix corresponding to the system.

The given system is inconsistent and there is no solution.

�2R1 � R2 → �10

�20

��

�413�

�2.5R1 → �1

2�2�4

��

�45�

��0.42

0.8�4

��

1.65�

A�1 does not exist.

A�1 �1

1.6 � 1.6 ��4

�2

�0.8

�0.4�

A � ��0.4

2

0.8

�4� 56.

Solution: �6, �2�

� �1

0.32��1.92

0.64� � � 6

�2�

�x

y� � �1

0.32�1.4

1

0.6

0.2��2.4

�8.8�

A�1 �1

0.28 � 0.6�1.4

1

0.6

0.2�

A � � 0.2

�1

�0.6

1.4�

Section 8.3 The Inverse of a Square Matrix 759

57.

Solution: ��4, �8�

�xy� � ��1

2

1213� � �2

�12� � ��4�8�

A�1 �1

�3

16 �916�

34

�32

�38

�14� � �

43�

34

�32

�38

�14� � ��1

2

1213�

A � ��1432

3834� 58.

Solution: ��12, 10�

� �1219� 19

�956� � ��12

10� �x

y� � �1219 ��

72

�43

156���20

�51�

A�1 �1

�3512 �

43��

72

�43

156�

A � �56

43

�1

�72�

59.

Find

Solution: ��1, 3, 2�

�x

y

z� �

155 �

18

3

�14

4

19

3

�5

�10

10� �

�5

10

1� �

155 �

�55

165

110� � �

�1

3

2�

A�1 �155�

18

3

�14

4

19

3

�5

�10

10�

�17R3 � R1 →

�12R3 � R2 → �1

0

0

0

1

0

0

0

1

���

1855355

�1455

45519553

55

�1

11

�2

112

11� � �I � A�1�

�1

55R3 → �

100

010

17121

���

�4�3�

1455

11355

322

11�

R2 � R1 →

�3R2 � R3 → �

100

010

1712

�55

���

�4�314

11

�3

32

�10�

�R3 � R2 → �100

�113

512

�19

���

�1�3

5

010

12

�4�

�2R1 � R2 →�4R1 � R3 →

�100

�143

5�7

�19

���

�125

010

1�2�4�

�R3 � R1 →

�124

�12

�1

531

���

�101

010

100�

R1

R3

�524

�22

�1

631

���

001

010

100�

�A � I� � �425

�12

�2

136

���

100

010

001�

A�1.

A � �4

2

5

�1

2

�2

1

3

6�

760 Chapter 8 Matrices and Determinants

61.

does not exist. This implies that there is no uniquesolution; that is, either the system is inconsistent or the system has infinitely many solutions. Use a graphing utilityto find the reduced row-echelon form of the matrix corresponding to the system.

Let Then and

Solution: where a is a real number� 516a �

1316, 19

16a �1116, a�

y �1916a �

1116.x �

516a �

1316z � a.

�x

y

516z1916z

13161116

�1

0

0

0

1

0

�5

16

�1916

0

13161116

0�

�521

�32

�7

2�3

8

���

23

�4�

A�1

A � �521

�32

�7

2�3

8�

62.

does not exist. This implies that there is no uniquesolution; that is, either the system is inconsistent orthe system has infinitely many solutions. Use a graphing utility to find the reduced row-echelon form of thematrix corresponding to the system.

Let Then and

Solution: where a is a real number�2a � 1, �3a � 2, a�

y � �3a � 2.x � 2a � 1z � a.

�x

y

2z

3z

�1

2

�100

010

�230

���

�120�

�235

359

59

17

���

47

13�

A�1

A � �235

359

59

17� 63.

Solution: ��7, 3, �2�

�x

y

z� � �

0.56

�0.04

�0.76

0.12

�0.08

�0.52

�0.2

�0.2

0.2��

�29

37

�24� � �

�7

3

�2�

A�1 � �0.56

�0.04

�0.76

0.12

�0.08

�0.52

�0.2

�0.2

0.2�

A � �3

�4

1

�2

1

�5

1

�3

1�

64.

Solution: �10, �3, 5�

�x

y

z� � �

� 0.034

�0.086

�0.133

0.066

0.117

0.029

�0.004

�0.139

�0.094��

�151

86

187� � �

10

�3

5�

A�1 � ��0.034

�0.086

�0.133

0.066

0.117

0.029

�0.004

�0.139

�0.094�

A � ��8

12

15

7

3

�9

�10

�5

2� 65.

Solution: �5, 0, �2, 3�

�x

y

z

w� � �

0

�1

�2

�1

�1

�5

�4

�4

0

0

1

0

1

3

�2

1� �

41

�13

12

�8� � �

5

0

�2

3�

A�1 � �0

�1

�2

�1

�1

�5

�4

�4

0

0

1

0

1

3

�2

1�

A � �7

�2

4

�1

�3

1

0

1

0

0

1

0

2

�1

�2

�1�

60.

Solution: �5, 8, �2�

�x

y

z� �

1

82 �

�21

�44

26

19

32

�4

16

14

�12��

�2

16

4� �

1

82 �410

656

�164� � �

5

8

�2�

A�1 �1

82 �

�21

�44

26

19

32

�4

16

14

�12�

A � �4

2

8

�2

2

�5

3

5

�2�

Section 8.3 The Inverse of a Square Matrix 761

66.

Solution: �6.21, �0.77, �2.67, 2.40�

�x

y

z

w� � �

0.338

0.042

�0.141

0.113

�0.352

0.164

0.230

�0.117

0.141

�0.066

0.108

0.047

0.394

�0.117

�0.164

�0.202��

11

�7

3

�1� � �

6.21

�0.77

�2.67

2.40�

A�1 � �0.338

0.042

�0.141

0.113

�0.352

0.164

0.230

�0.117

0.141

�0.066

0.108

0.047

0.394

�0.117

�0.164

�0.202�

A � �2

1

2

1

5

4

�2

0

0

2

5

0

1

�2

1

�3�

67.

Solution: $7000 in AAA-rated bonds, $1000 in A-rated bonds, $2000 in B-rated bonds

X � A�1B �111�

50

�13

�26

�600

200

400

�4

5

�1� �

10,000

705

0� � �

7000

1000

2000�

4R3 � R1 →

�5R3 � R2 → �1

0

0

0

1

0

0

0

1

5011

�1311

�2611

�60011

20011

40011

�4

115

11

�1

11

� � �I � A�1�

�111R3 →

�100

010

�451

���

14�13�

2611

�200200

40011

00

�111�

�R2 � R1 →

�2R2 � R3 → �

100

010

�45

�11

���

14�13

26

�200200

�400

001�

�13R1 � R2 → �100

112

15

�1

���

1�13

0

0200

0

001�

200R2 → �1

130

1142

118

�1

���

1 0 0

0200

0

001�

�A � I� � �1

0.065

0

1

0.07

2

1

0.09

�1

���

1

0

0

0

1

0

0

0

1�

A � �1

0.065

0

1

0.07

2

1

0.09

�1�

762 Chapter 8 Matrices and Determinants

69. Use the inverse matrix from Exercise 67.

Solution: $9000 in AAA-rated bonds, $1000 in A-rated bonds, $2000 in B-rated bonds

X � A�1B �111�

50

�13

�26

�600

200

400

�4

5

�1� �

12,000

835

0� � �

9000

1000

2000�

A�1

70. Use the inverse matrix from Exercise 69.

Solution: $200,000 in AAA-rated bonds, $100,000 in A-rated bonds, and $200,000 in B-rated bonds.

X � A�1B �1

11 �

50

�13

�26

�600

200

400

�4

5

�1��

500,000

38,000

0� � �

200,000

100,000

200,000�

A�1

68.

Solution: $4000 in AAA-rated bonds, $2000 in A-rated bonds, $4000 in B-rated bonds.

X � A�1B �1

11 �

50

�13

�26

�600

200

400

�4

5

�1��

10,000

760

0� � �

4000

2000

4000�

4R3 � R1 →

�5R3 � R2 → �1

0

0

0

1

0

0

0

1

5011

�1311

�2611

�600112001140011

�4

115

11

�1

11

� � �I � A�1�

�111R3 →

�100

010

�451

���

14�13�

2611

�200200

40011

00

�111�

�R2 � R1 →

�2R2 � R3 → �

1

0

0

0

1

0

�4

5

�11

���

14

�13

26

�200

200

�400

0

0

1�

�13R1 � R2 →

�100

112

15

�1

���

1�13

0

0200

0

001�

200R2 → �1

13

0

1

14

2

1

18

�1

���

1

0

0

0

200

0

0

0

1�

�A � I� � �1

0.065

0

1

0.07

2

1

0.09

�1

���

1

0

0

0

1

0

0

0

1�

A � �1

0.065

0

1

0.07

2

1

0.09

�1�

Section 8.3 The Inverse of a Square Matrix 763

73. True. If B is the inverse of A, then AB � I � BA. 74. True. If A and B are both square matrices and it can be shown that BA � In .

AB � In,

71. (a)

Solution: I1 � �3 amperes, I2 � 8 amperes, I3 � 5 amperes

�I1

I2

I3� �

114 �

5

�4

1

�4

6

2

4

8

�2� �

14

28

0� � �

�3

8

5�

A�1 �114 �

5

�4

1

�4

6

2

4

8

�2�

5R3 � R1 →

�4R3 � R2 → �1

0

0

0

1

0

0

0

1

514

�271

14

�273717

2747

�17

� � �I � A�1�

114R3 →

�100

010

�541

���

00114

�1117

10

�17�

�R2 � R1 →

2R2 � R3 → �

100

010

�54

14

���

001

�112

10

�2�

�2R1 � R3 → �

100

11

�2

�146

���

001

010

10

�2�

R1

R3

�102

110

�144

���

001

010

100�

�A � I� � �201

011

44

�1

���

100

010

001�

A � �2

0

1

0

1

1

4

4

�1�

72. (a)

System:

(e)

Since represents 2008, the model projects thatthe number of licensed drivers will reach 208 millionduring 2008.

t � 18

t � 18.7

2.15t � 40.3

2.15t � 167.7 � 208

� 3b � 27a � 561.227b � 251a � 5068

�n

i�1xi yi � 7�182.7� � 9�187.2� � 11�191.3� � 5068

�n

i�1xi

2 � 49 � 81 � 121 � 251

�n

i�1yi � 182.7 � 187.2 � 191.3 � 561.2;

n � 3; �n

i�1xi � 7 � 9 � 11 � 27;

(b)

The least squares regression line is

(c) For 2003,

This projects about 196 million licensed drivers in 2003.

(d) The projected value is very close to the actual value.

t � 13; y � 2.15�13� � 167.7 � 195.65.

y � 2.15t � 167.7.

b � 167.7, a � 2.15

� � �25124 ��561.2� � ��9

8��5068�

��98��561.2� � �1

8��5068� � � �167.72.15�

� 327

27251�

�1� �

25124

�98

�9818�; �

25124

�98

�9818��561.2

5068�

(b)

Solution:

I1 � 2 amperes, I2 � 3 amperes, I3 � 5 amperes

�I1

I2

I3� �

114 �

5

�4

1

�4

6

2

4

8

�2� �

24

23

0� � �

2

3

5�

764 Chapter 8 Matrices and Determinants

75.

A�1A �1

ad � bc � d

�c

�b

a��a

c

b

d� �1

ad � bc �ad � bc

0

0

ad � bc� � �1

0

0

1�

�1

ad � bc �ad � bc

0

0

ad � bc� � �1

0

0

1�

AA�1 � �a

c

b

d� 1

ad � bc�d

�c

�b

a� �1

ad � bc �a

c

b

d��d

�c

�b

a�

76. (a)

Given A � �a11

0

0

0

a22

0

0

0

a33�, A�1 � � 0

0

0

0

0

0 � .

Given A � �a11

0

0

a22�, A�1 � �

0

0

�.

1a11

1a22

1a11 1

a22 1a33

77.

x

−9 −8 −7 −6 −5 −4−10

x ≤ �9 or x ≥ �5

x � 7 ≤ �2 or x � 7 ≥ 2

x � 7 ≥ 2 78.

x

−1 0 1 2 3 4−2

�1 < x < 2

�2 < 2x < 4

�3 < 2x � 1 < 3

2x � 1 < 3 79.

x �2 ln 315

ln 3� 10.472

x

2 ln 3 � ln 315

ln 3x�2 � ln 315

3x�2 � 315

80.

x � �5 ln 1

5� �8.047

�x

5� ln

1

5

ln e�x�5 � ln 15

e�x�5 �1

5

2000e�x�5 � 400 81.

x � 26.5 � 90.510

log2 x � 6.5

log2 x � 2 � 4.5 82.

Choose the positive value only:

x �1 � �5

2� 1.618

x �1 ± �5

2

x �1 ± �1 � 4��1�

2

x2 � x � 1 � 0

x�x � 1� � 1

eln�x�x�1�� � e0

ln�x�x � 1�� � 0

ln x � ln�x � 1� � 0

(b) In general, the inverse of a matrix in the form of A is

� 0

0

..

.

0

0

0

..

.

0

0

0

..

.

0

. . .

. . .

. . .

. . .

. . .

0

0

0

..

.

� 1

a111

a22

1a33

1ann

83. Answers will vary.

Section 8.4 The Determinant of a Square Matrix

Section 8.4 The Determinant of a Square Matrix 765

■ You should be able to determine the determinant of a matrix of order by using the difference of the productsof the diagonals.

■ You should be able to use expansion by cofactors to find the determinant of a matrix of order or greater.

■ The determinant of a triangular matrix equals the product of the entries on the main diagonal.

3 � 3

2 � 2

1. 5 2. �8 3. �23 1

4� � 2�4� � 1�3� � 8 � 3 � 5

5. � 5

�6

2

3� � 5�3� � 2��6� � 15 � 12 � 274. ��3

5

1

2� � ��3��2� � �5��1� � �11

6. �24 �2

3� � �2��3� � �4���2� � 14 7. ��73

00� � �7�0� � 0�3� � 0

8. �40 �3

0� � �4��0� � �0���3� � 0 9. �20 6

3� � 2�3� � 6�0� � 6

10. � 2

�6

�3

9� � �2��9� � ��6���3� � 0 11. ��3�6

�2�1� � ��3���1� � ��2���6� � 3 � 12 � �9

12. � 4

�2

7

5� � �4��5� � ��2��7� � 34 13. �97 08� � 9�8� � 0�7� � 72 � 0 � 72

16. � 23

�1

43

�13 � � �2

3���13� � ��1��4

3� �109

14. � 0

�3

6

2� � �0��2� � ��3��6� � 18 15. � �12

�6

1313� � �

12�1

3� �13��6� � �

16 � 2 �

116

17. � 0.3

0.2

�0.4

0.2

0.2

0.4

0.2

0.2

0.3� � �0.002

19. � 0.9�0.1�2.2

0.70.34.2

01.36.1� � �4.84218. � 0.1

�0.3

0.5

0.2

0.2

0.4

0.3

0.2

0.4� � �0.022 20. �0.1

7.5

0.3

0.1

6.2

0.6

�4.3

0.7

�1.2� � �11.217

21. � 1

3

�2

4

6

1

�2

�6

4� � 0 22. �200 3

5

0

1

�2

�2� � �20

Vocabulary Check

1. determinant 2. minor 3. cofactor 4. expanding by cofactors

766 Chapter 8 Matrices and Determinants

23.

(a) (b)

C22 � M22 � 3M22 � 3

C21 � �M21 � �4M21 � 4

C12 � �M12 � �2M12 � 2

C11 � M11 � �5M11 � �5

�3

2

4

�5� 24.

(a)

M22 � 11

M21 � 0

M12 � �3

M11 � 2

� 11

�3

0

2�(b)

C22 � M22 � 11

C21 � M21 � 0

C12 � �M12 � 3

C11 � M11 � 2

25.

(a) (b)

C22 � M22 � 3M22 � 3

C21 � �M21 � �1M21 � 1

C12 � �M12 � 2M12 � �2

C11 � M11 � �4M11 � �4

� 3

�2

1

�4� 26.

(a)

M22 � �6

M21 � 5

M12 � 7

M11 � �2

��6

7

5

�2�(b)

C22 � M22 � �6

C21 � �M21 � �5

C12 � �M12 � �7

C11 � M11 � �2

27.

(a)

(b)

C33 � ��1�6M33 � 8

C32 � ��1�5M32 � �10

C31 � ��1�4M31 � �4

C23 � ��1�5M23 � 4

C22 � ��1�4M22 � 2

C21 � ��1�3M21 � �2

C13 � ��1�4M13 � 1

C12 � ��1�3M12 � 4

C11 � ��1�2M11 � 3

M33 � � 4

�3

0

2� � 8 � 0 � 8

M32 � � 4

�3

2

1� � 4 � ��6� � 10

M31 � �02 2

1� � 0 � 4 � �4

M23 � �41 0

�1� � �4 � 0 � �4

M22 � �41 2

1� � 4 � 2 � 2

M21 � � 0

�1

2

1� � 0 � ��2� � 2

M13 � ��3

1

2

�1� � 3 � 2 � 1

M12 � ��3

1

1

1� � �3 � 1 � �4

M11 � � 2

�1

1

1� � 2 � ��1� � 3

�4

�3

1

0

2

�1

2

1

1� 28.

(a)

(b)

C33 � ��1�6M33 � 5

C32 � ��1�5M32 � �5

C31 � ��1�4M31 � �5

C23 � ��1�5M23 � 2

C22 � ��1�4M22 � 4

C21 � ��1�3M21 � 4

C13 � ��1�4M13 � �26

C12 � ��1�3M12 � 8

C11 � ��1�2M11 � 38

M33 � �13 �1

2� � 2 � ��3� � 5

M32 � �13 0

5� � 5 � 0 � 5

M31 � ��1

2

0

5� � �5 � 0 � �5

M23 � �14 �1

�6� � �6 � ��4� � �2

M22 � �14 0

4� � 4 � 0 � 4

M21 � ��1

�6

0

4� � �4 � 0 � �4

M13 � �34 2

�6� � �18 � 8 � �26

M12 � �34 5

4� � 12 � 20 � �8

M11 � � 2

�6

5

4� � 8���30� � 38

�1

3

4

�1

2

�6

0

5

4�

Section 8.4 The Determinant of a Square Matrix 767

29.

(a)

(b)

C33 � ��1�6M33 � 12

C32 � ��1�5M32 � 42

C31 � ��1�4M31 � �4

C23 � ��1�5M23 � �7

C22 � ��1�4M22 � 26

C21 � ��1�3M21 � 36

C13 � ��1�4M13 � 11

C12 � ��1�3M12 � �12

C11 � ��1�2M11 � 30

M33 � �33 �2

2� � 6 � 6 � 12

M32 � �33 8

�6� � �18 � 24 � �42

M31 � ��2

2

8

�6� � 12 � 16 � �4

M23 � � 3

�1

�2

3� � 9 � 2 � 7

M22 � � 3

�1

8

6� � 18 � 8 � 26

M21 � ��2

3

8

6� � �12 � 24 � �36

M13 � � 3

�1

2

3� � 9 � 2 � 11

M12 � � 3

�1

�6

6� � 18 � 6 � 12

M11 � �23 �6

6� � 12 � 18 � 30

�3

3

�1

�2

2

3

8

�6

6� 30.

(a)

(b)

C33 � ��1�6M33 � �51

C32 � ��1�5M32 � 28

C31 � ��1�4M31 � 24

C23 � ��1�5M23 � 68

C22 � ��1�4M22 � �12

C21 � ��1�3M21 � 82

C13 � ��1�4M13 � 85

C12 � ��1�3M12 � 42

C11 � ��1�2M11 � 36

M33 � ��2

7

9

�6� � �51

M32 � ��2

7

4

0� � �28

M31 � � 9

�6

4

0� � 24

M23 � ��2

6

9

7� � �68

M22 � ��2

6

4

�6� � �12

M21 � �97 4

�6� � �82

M13 � �76 �6

7� � 85

M12 � �76 0

�6� � �42

M11 � ��6

7

0

�6� � 36

��2

7

6

9

�6

7

4

0

�6�

31. (a)

(b) ��3

4

2

2

5

�3

1

6

1� � �2�42 6

1� � 5��3

2

1

1� � 3��3

4

1

6� � �2��8� � 5��5� � 3��22� � �75

��3

4

2

2

5

�3

1

6

1� � �3� 5

�3

6

1� � 2�42 6

1� � �42 5

�3� � �3�23� � 2��8� � 22 � �75

32. (a)

(b) ��3

6

4

4

3

�7

2

1

�8� � 2�64 3

�7� � ��3

4

4

�7� � 8��3

6

4

3� � 2��54� � �5� � 8��33� � 151

��3

6

4

4

3

�7

2

1

�8� � �6� 4

�7

2

�8� � 3��3

4

2

�8� � 1��3

4

4

�7� � �6��18� � 3�16� � �5� � 151

768 Chapter 8 Matrices and Determinants

33. (a)

(b) �501 0

12

6

�3

4

3� � 0�01 4

3� � 12�51 �3

3� � 6�50 �3

4� � 0��4� � 12�18� � 6�20� � 96

�501 0

12

6

�3

4

3� � 0�06 �3

3� � 12�51 �3

3� � 4�51 0

6� � 0�18� � 12�18� � 4�30� � 96

34. (a)

(b) �10

30

0

�5

0

10

5

10

1� � 10� 0

10

10

1� � 30��5

10

5

1� � 0��5

0

5

10� � 10��100� � 30��55� � 0��50� � 650

�10

30

0

�5

0

10

5

10

1� � 0��5

0

5

10� � 10�10

30

5

10� � �10

30

�5

0� � 0��50� � 10��50� � 150 � 650

35. (a)

(b)

� 0 � 13��298� � 0 � 6�674� � 170

� 6

4

�1

8

0

13

0

6

�3

6

7

0

5

�8

4

2� � 0� 4

�1

8

6

7

0

�8

4

2� � 13� 6

�1

8

�3

7

0

5

4

2� � 0�648 �3

6

0

5

�8

2� � 6� 6

4

�1

�3

6

7

5

�8

4� � �4��282� � 13��298� � 6��174� � 8��234� � 170

� 6

4

�1

8

0

13

0

6

�3

6

7

0

5

�8

4

2� � �4�006 �3

7

0

5

4

2� � 13� 6

�1

8

�3

7

0

5

4

2� � 6� 6

�1

8

0

0

6

5

4

2� � 8� 6

�1

8

0

0

6

�3

7

0�

36. (a)

(b)

� 10�24� � 4�245� � 0��64� � 1�427� � �1167

�10

4

0

1

8

0

3

0

3

5

2

�3

�7

�6

7

2� � 10�030 5

2

�3

�6

7

2� � 4�830 3

2

�3

�7

7

2� � 0�800 3

5

�3

�7

�6

2� � 1�803 3

5

2

�7

�6

7� � 0��64� � 3��3� � 2��112� � 7�136� � �1167

�10

4

0

1

8

0

3

0

3

5

2

�3

�7

�6

7

2� � 0�800 3

5

�3

�7

�6

2� � 3�10

4

1

3

5

�3

�7

�6

2� � 2�10

4

1

8

0

0

�7

�6

2� � 7�10

4

1

8

0

0

3

5

�3�

37. Expand along Column 1.

�244 �1

2

2

0

1

1� � 2�22 1

1� � 4��1

2

0

1� � 4��1

2

0

1� � 2�0� � 4��1� � 4��1� � 0

38. Expand along Row 3.

� 0�3� � 1��3� � 4�0� � 3

��2

1

0

2

�1

1

3

0

4� � 0� 2

�1

3

0� � 1��2

1

3

0� � 4��2

1

2

�1�

Section 8.4 The Determinant of a Square Matrix 769

39. Expand along Row 2.

�604 3

0

�6

�7

0

3� � 0� 3

�6

�7

3� � 0�64 �7

3� � 0�64 3

�6� � 0

40. Expand along Column 3.

� 2�2� � 0�2� � 3��2� � �2

� 1

3

�2

1

1

0

2

0

3� � 2� 3

�2

1

0� � 0� 1

�2

1

0� � 3�13 1

1�41. (Upper triangular)��1

0

0

2

3

0

�5

4

3� � ��1��3��3� � �9

42. Expand along Row 1.

� 1��5� � 0��20� � 0�1� � �5

� 1

�4

5

0

�1

1

0

0

5� � 1��1

1

0

5� � 0��4

5

0

5� � 0��4

5

�1

1�43. Expand along Column 3.

� �2�14� � 3��10� � �58

� 1

3

�1

4

2

4

�2

0

3� � �2� 3

�1

2

4� � 3�13 4

2�

44. Expand along Row 3.

�211 �1

4

0

3

4

2� � 1��1

4

3

4� � 0�21 3

4� � 2�21 �1

4� � 1��16� � 0�5� � 2�9� � 2

45. (Upper triangular)�200 4

3

0

6

1

�5� � �2��3���5� � �30 46. Expand along Row 1.

� �3�22� � 0�14� � 0�3� � �66

��3

7

1

0

11

2

0

0

2� � �3�11

2

0

2� � 0�71 0

2� � 0�71 11

2�

47. Expand along Column 3.

�2213 6

7

5

7

6

3

0

0

2

6

1

7� � 6�213 7

5

7

6

1

7� � 3�213 6

5

7

2

1

7� � 6��20� � 3�16� � �168

48. Expand along Row 2.

� 3

�2

1

0

6

0

1

3

�5

6

2

�1

4

0

2

�1� � ���2��613 �5

2

�1

4

2

�1� � 6�310 6

1

3

4

2

�1� � 2��63� � 6��3� � �108

49. Expand along Column 1.

�5400 3

6

2

1

0

4

�3

�2

6

12

4

2� � 5�621 4

�3

�2

12

4

2� � 4�321 0

�3

�2

6

4

2� � 5�0� � 4�0� � 0

770 Chapter 8 Matrices and Determinants

50. Expand along Row 3.

� 1

�5

0

3

4

6

0

�2

3

2

0

1

2

1

0

5� � 0

51. Expand along Column 2, then along Column 4.

�� 3

�2

1

6

3

2

0

0

0

0

4

1

0

2

5

�1

3

4

�1

1

5

2

0

0

0�� � �2��2

1

6

3

1

0

2

5

3

4

�1

1

2

0

0

0� � ��2���2��163 0

2

5

4

�1

1� � 4�103� � 412

52. Expand along Column 1.

5

0

0

0

0

2

1

0

0

0

0

4

2

3

0

0

3

6

4

0

�2

2

3

1

2

� � 5�1000 4

2

3

0

3

6

4

0

2

3

1

2� � 5 � 1�230 6

4

0

3

1

2� � 5��20� � �100

53. �308 8

�5

1

�7

4

6� � �126 54. � 5

9

�8

�8

7

7

0

4

1� � 223 55. � 7

�2

�6

0

5

2

�14

4

12� � 0

56. � 3

�2

12

0

5

5

0

0

7� � 105 57. �1220 �1

6

0

2

8

0

2

8

4

�4

6

0� � �336 58. � 0

8

�4

�7

�3

1

6

0

8

�1

0

0

2

6

9

14� � 7441

59. �� 3

�1

5

4

1

�2

0

�1

7

2

4

2

0

�8

3

3

1

3

0

0

1

0

2

0

2�� � 410 60. �

�2

0

0

0

0

0

3

0

0

0

0

0

�1

0

0

0

0

0

2

0

0

0

0

0

�4� � �48

61. (a)

(b)

(c)

(d) ��2

0

0

�3� � 6

��1

0

0

3� �2

0

0

�1� � ��2

0

0

�3��20 0

�1� � �2

��1

0

0

3� � �3 62. (a)

(b)

(c)

(d) �AB� � ��2

4

�5

10� � 0

AB � ��2

4

1

�2��1

0

2

�1� � ��2

4

�5

10�

�B� � �10 2

�1� � �1

�A� � ��2

4

1

�2� � 0

Section 8.4 The Determinant of a Square Matrix 771

63. (a)

(b)

(c)

(d) ��41

4�1� � 0

�43

0�2� ��1

�212� � ��4

14

�1���1�2

12� � 0

�43 0�2� � �8 64. (a)

(b)

(c)

(d) �AB� � � 4

�1

22

20� � 102

AB � �5

3

4

�1��0

1

6

�2� � � 4

�1

22

20�

�B� � �01 6

�2� � �6

�A� � �53 4

�1� � �17

65. (a)

(b)

(c)

(d) � 7�8

7

19

�3

4�3

9� � 399

�0

�30

1�2

4

211� �

313

�2�1

1

021� � �

7�8

7

19

�3

4�3

9��313

�2�1

1

021� � �19

� 0�3

0

1�2

4

211� � �21 66. (a)

(b)

(c)

(d) �AB� � ��9

�5

4

4

�10

�1

1

6

�1� � �23

� ��9

�5

4

4

�10

�1

1

6

�1�

AB � �3

�1

�2

2

�3

0

0

4

1��

�3

0

�2

0

2

�1

1

�1

1�

�B� � ��3

0

�2

0

2

�1

1

�1

1� � 1

�A� � � 3

�1

�2

2

�3

0

0

4

1� � �23

67. (a)

(b)

(c)

(d) � 1

�1

0

4

0

2

3

3

0� � �12

��1

1

0

2

0

1

1

1

0� �

�1

0

0

0

2

0

0

0

3� � �

1

�1

0

4

0

2

3

3

0�

��1

0

0

0

2

0

0

0

3� � �6

��1

1

0

2

0

1

1

1

0� � 2 68. (a)

(b)

(c)

(d) �AB� � �786 �4

�6

�2

9

3

15� � 0

AB � �2

1

3

0

�1

1

1

2

0��

2

0

3

�1

1

�2

4

3

1� � �

7

8

6

�4

�6

�2

9

3

15�

�B� � �203 �1

1

�2

4

3

1� � �7

�A� � �213 0

�1

1

1

2

0� � 0

69.

Thus, �wy x

z� � �� y

w

z

x�. �� y

w

z

x� � ��xy � wz� � wz � xy

�wy x

z� � wz � xy 70.

So, �wy cx

cz� � c�wy x

z�.c�wy x

z� � c�wz � xy�

�wy cx

cz� � cwz � cxy � c�wz � xy�

772 Chapter 8 Matrices and Determinants

71.

Thus, �wy x

z� � �wy x � cw

z � cy�. �wy x � cw

z � cy� � w�z � cy� � y�x � cw� � wz � xy

�wy x

z� � wz � xy 72.

So, � w

cw

x

cx� � 0.

� w

cw

x

cx� � cxw � cxw � 0

73.

� �y � x��z � x��z � y�

� �y � x��z�z � x� � y�z � x��

� �y � x��z2 � zx � zy � xy�

� �y � x��z2 � zy � zx � xy�

� �y � x��z2 � z�y � x� � xy�

� z2�y � x� � z�y � x��y � x� � xy�y � x�

� z2�y � x� � z�y2 � x2� � xy�y � x�

� yz2 � xz2 � y2z � x2z � xy�y � x�

� �yz2 � y2z� � �xz2 � x2z� � �xy2 � x2y�

�111 x

y

z

x2

y2

z2� � �yz y2

z2� � �xz x2

z2� � �xy x2

y2�

74.

� 3ab2 � b3 � b2�3a � b�

� a3 � 3a2b � 3ab2 � b3 � 3a3 � 3a2b � 2a3

� �a � b�3 � 3a2�a � b� � 2a3

� �a � b�3 � a2�a � b� � a2�a � b� � a3 � a3 � a2�a � b�

� �a � b���a � b�2 � a2� � a�a�a � b� � a2� � a�a2 � a�a � b��

�a � b

a

a

a

a � b

a

a

a

a � b� � �a � b��a � b

a

a

a � b� � a�aa a

a � b� � a� a

a � b

a

a �

75.

x � �1 or x � 4

�x � 1��x � 4� � 0

x2 � 3x � 4 � 0

�x � 1��x � 2� � 6 � 0

�x � 1

3

2

x � 2� � 0 76.

x � �1 or x � 3

�x � 1��x � 3� � 0

x2 � 2x � 3 � 0

x�x � 2� � ��3���1� � 0

�x � 2

�3

�1

x� � 0

77.

x � �1 or x � �4

�x � 1��x � 4� � 0

x2 � 5x � 4 � 0

�x � 3��x � 2� � 2 � 0

�x � 3

1

2

x � 2� � 0 78.

x � �2 or x � 3

�x � 2��x � 3� � 0

x2 � x � 6 � 0

�x � 4��x � 5� � 7��2� � 0

�x � 4

7

�2

x � 5� � 0

Section 8.4 The Determinant of a Square Matrix 773

79. � 4u

�1

�1

2v� � 8uv � 1

81. � e2x

2e2x

e3x

3e3x � � 3e5x � 2e5x � e5x

80. �3x2

1

�3y2

1 � � 3x2 � ��3y2� � 3x2 � 3y2

82. � e�x

�e�x

xe�x

�1 � x�e�x � � �1 � x�e�2x � ��xe�2x� � e�2x � xe�2x � xe�2x � e�2x

83. �x1 ln x

1x � � 1 � ln x 84.

� x � x ln x � x ln x � x

�x

1

x ln x

1 � ln x� � x�1 � ln x� � x ln x

85. True. If an entire row is zero, then each cofactorin the expansion is multiplied by zero.

86. True. If a square matrix has two columns that are equal,then elementary column operations can be used to create acolumn with all zeros.

87. Let

Thus, Your answer may differ, depending on how you choose and B.A�A � B� � �A� � �B�.

A � B � ��3

1

3

9�, �A � B� � ��3

1

3

9� � �30

�A� � � 1

�2

3

4� � 10, �B� � ��4

3

0

5� � �20, �A� � �B� � �10

A � � 1

�2

3

4� and B � ��4

3

0

5�.

88. (a)

For an matrix with consecutive integer entries, the determinant appears to be 0.

(b)

� �3x � 6x � 6 � 3x � 6 � 0

� �3x � �x � 1���6� � �x � 2���3�

� �x2 � 11x � 30�� � �x � 2���x2 � 10x � 21� � �x2 � 10x � 24��

� x��x2 � 12x � 32� � �x2 � 12x � 35�� � �x � 1���x2 � 11x � 24�

� �x � 6��x � 5�� � �x � 2���x � 3��x � 7� � �x � 6��x � 4��

� x��x � 4��x � 8� � �x � 7 ��x � 5�� � �x � 1���x � 3��x � 8�

� x

x � 3

x � 6

x � 1

x � 4

x � 7

x � 2

x � 5

x � 8� � x�x � 4

x � 7

x � 5

x � 8� � �x � 1��x � 3

x � 6

x � 5

x � 8� � �x � 2��x � 3

x � 6

x � 4

x � 7��n > 2�n � n

�57

61

65

69

58

62

66

70

59

63

67

71

60

64

68

72� � 0�19

23

27

31

20

24

28

32

21

25

29

33

22

26

30

34� � 0

��5

�2

1

�4

�1

2

�3

0

3� � 0�33

36

39

34

37

40

35

38

41� � 0

� 4

7

10

5

8

11

6

9

12� � 0

774 Chapter 8 Matrices and Determinants

89. A square matrix is a square array of numbers. The determinant of a square matrix is a real number.

90.

So, �2A� � 8�A� � 8�5� � 40.

� 8�A� � 8�x11�x22 x33 � x32 x23� � x12�x21 x33 � x31 x23� � x13�x21 x32 � x31 x22��

� 2�x11�4x22 x33 � 4x32 x23� � x12�4x21 x33 � 4x31 x23� � x13�4x21 x32 � 4x31 x22��

�2A� � 2x11�2x22

2x32

2x23

2x33� � 2x12�2x21

2x31

2x23

2x33� � 2x13�2x21

2x31

2x22

2x32� 2A � �

2x11

2x21

2x31

2x12

2x22

2x32

2x13

2x23

2x33�

Let A � �x11

x21

x31

x12

x22

x32

x13

x23

x33� and �A� � 5.

91. (a)

Column 2 and Column 3 were interchanged.

�� 1�7

6

4�5

2

321� � �115

� 1�7

6

321

4�5

2� � �115 (b)

Row 1 and Row 3 were interchanged.

�� 1�2

1

623

204� � �40

� 1�2

1

326

402� � �40

92. (a) Multiplying Row 1 of the matrix by and adding it to Row 2 gives the matrix

(b) Multiplying Row 2 of the matrix by and adding it to Row 1 gives the matrix

�527 4�3

6

243� � �11 � �127 10

�36

�643�

�127

10�3

6

�643�.�2�

527

4�3

6

243�

�15 �32� � 17 � �10 �3

17��1

0�317�.�5�1

5�3

2�

93. (a)

Row 1 was multiplied by 5.

�B� � 5�A�

5�A� � 5�12 2�3� � �35

�B� � �52 10�3� � �35

A � �12

2�3�, B � �5

210

�3� (b)

Column 2 was multiplied by 4 and Column 3 was multiplied by 3.

�B� � �4��3��A� � 12�A�

12�A� � 12�137 2�3

1

�123� � �300

�B� � �137

8�12

4

�369� � �300

A � �137

2�3

1

�123�, B � �

137

8�12

4

�369�

Section 8.4 The Determinant of a Square Matrix 775

94. (a)

(b)

(c)

Using cofactors and

In each case, the determinant of the matrix is the product of the diagonal entries. From this, one would conjecture that the determinant of a diagonal matrix is the product of the diagonal entries.

�A� � 2C11 � 2��2 � 1 � 3� � 2 � ��6� � �12

C11 � ��200

010

003�

�A� � 2 � C11 � 0 � C12 � 0 � C13 � 0 � C14.a11,

A � �2000 0�2

00

0010

0003�

A � ��100

050

002�, �A� � ��1��5��2� � �10

A � �70 04�, �A� � 7�4� � 0 � 28

95.

Since f is a polynomial, the domain is all real numbers x.

f �x� � x3 � 2x 96.

An odd root of a number is defined for all real numbers.Domain: all real numbers x

g�x� � 3�x

97.

Critical numbers:

Test intervals:

Test: Is

Solution:

Domain of h: �4 ≤ x ≤ 4

��4, 4�

16 � x2 ≥ 0?

���, �4�, ��4, 4�, �4, ��

x � ±4

�4 � x��4 � x� ≥ 0

16 � x2 ≥ 0

h�x� � �16 � x2 98.

Domain: all real numbers x � ±6

36 � x2 � 0 ⇒ x2 � 36 ⇒ x � ±6

A�x� �3

36 � x2

99.

Domain: all real numbers t > 1

t > 1

t � 1 > 0

g�t� � ln�t � 1� 100.

The exponential function is defined for all real numbers.

Domain: all real numbers

y � Aex

f �s� � 625e�0.5S

101.

x

4

12

4 8 12−4−8

y

x

2x

y

x

y

≤≥<

8

�3

5

102.

x

2

2−2−4

−4

−6

y

776 Chapter 8 Matrices and Determinants

103.

A�1 � �14

2

14

1�

14R1 →

�1

0

0

1

14

2

14

1� � �I � A�1�

�R2 � R1 →

��40

01

��

�12

�11�

2R1 � R2 → ��40

11

��

12

01�

�A � I� � ��48

1�1

��

10

01� 104.

A�1 � ��112

�43

56�

�2R2 � R1 →

�1

0

0

1��

�112

�4356� � �I � A�1�

12R2 →

�1

0

2

1

012

13

56�

5R1 � R2 → �1

0

2

2

0

1

13

53�

13R1 → � 1

�5

2

�8��

0

1

13

0� R2

R1

� 3

�5

6

�8��

0

1

1

0�

�A � I� � ��5

3

�8

6��

1

0

0

1�

105.

The zeros in Row 3 imply that the inverse does not exist.

�4724R2 � R3 →

�1

0

0

�14

24

0

�15

24

0

1

�21112

4

�74124

0

0

1�

�2R1 � R2 →�3R1 � R3 →

�100

�142447

�152447

���

1�2�3

4�7

�12

001�

4R2 � R1 →

�123

�14�4

5

�15�6

2

���

100

410

001�

�A � I� � ��7

23

2�4

5

9�6

2

���

100

010

001�

106.

—CONTINUED—

16R2 →

1

0

0

3

1

20

�2

�12

�12

0

0

1

113

6

016

0�

2R1 � R2 →6R1 � R3 →

�1

0

0

3

6

20

�2

�3

�12

���

0

0

1

1

2

6

0

1

0�

R3

R2

�1

�2�6

302

�210

���

001

100

010�

R2

R1 �1

�6

�2

3

2

0

�2

0

1

���

0

1

0

1

0

0

0

0

1�

�A � I� � ��6

1

�2

2

3

0

0

�2

1

���

1

0

0

0

1

0

0

0

1�

Section 8.5 Applications of Matrices and Determinants 777

106. —CONTINUED—

A�1 � ��

14

�14

�12

16

12

13

13

153

12R3 � R1 →12R3 � R2 → �

1

0

0

0

1

0

0

0

1

�14

�14

�12

16

12

13

13

153

� � �I � A�1�

�3R2 � R1 →

�1

0

0

0

1

0

�12

�12

1

0

0

�12

013

13

�12

16

53

�12R3 →

�1

0

0

3

1

0

�2

�12

1

0

0

�12

113

13

016

53

�20R2 � R3 →

�1

0

0

3

1

0

�2

�12

�2

0

0

1

113

�23

016

�103

Section 8.5 Applications of Matrices and Determinants

■ You should be able to use Cramer’s Rule to solve a system of linear equations.

■ Now you should be able to solve a system of linear equations by graphing, substitution, elimination, elementary row operations on an augmented matrix, using the inverse matrix, or Cramer’s Rule.

■ You should be able to find the area of a triangle with vertices

The symbol indicates that the appropriate sign should be chosen so that the area is positive.

■ You should be able to test to see if three points, are collinear.

if and only if they are collinear.

■ You should be able to find the equation of the line through by evaluating.

■ You should be able to encode and decode messages by using an invertible matrix.n � n

� x

x1

x2

y

y1

y2

1

1

1� � 0

�x1, y1� and �x2, y2��x1

x2

x3

y1

y2

y3

1

1

1� � 0,

�x1, y1�, �x2, y2�, and �x3, y3�,±

Area � ±12�x1

x2

x3

y1

y2

y3

1

1

1��x1, y1�, �x2, y2�, and �x3, y3�.

Vocabulary Check

1. Cramer’s Rule 2. colinear 3. 4. cryptogram 5. uncoded; codedA � ±12�x1

x2

x3

y1

y2

y3

111�

778 Chapter 8 Matrices and Determinants

1.

Solution: �2, �2�

y ��35 �2

4��35 4

3� �22

�11� �2

x ���2

4

4

3��35 4

3� ��22

�11� 2

�3x

5x

4y

3y

�2

4

2.

Solution: ��3, �5�

y ���4�1

47�27�

��4�1

�76� �

155�31

� �5

x �� 47�27

�76�

��4�1

�76� �

93�31

� �3

��4x

�x

7y

6y

47

�27

3.

Since Cramer’s Rule does not apply.

The system is inconsistent in this case and has no solution.

�36 24� � 0,

�3x

6x

2y

4y

�2

44.

Solution: �7, 5�

y �� 6�13

17�76�

� 6�13

�53� �

�235�47

� 5

x �� 17�76

�53�

� 6�13

�53� �

�329�47

� 7

� 6x

�13x

5y

3y

17

�76

5.

Solution: 327

, 307

y ���0.4

0.2

1.6

2.2���0.4

0.2

0.8

0.3� ��1.20

�0.28�

30

7

x ��1.6

2.2

0.8

0.3���0.4

0.2

0.8

0.3� ��1.28

�0.28�

32

7

��0.4x

0.2x

0.8y

0.3y

1.6

2.26.

Solution: 85

, �8310

y �� 2.4�4.6

14.63�11.51�

� 2.4�4.6

�1.30.5� �

39.674�4.78

��8310

x �� 14.63�11.51

�1.30.5�

� 2.4�4.6

�1.30.5� �

�7.648�4.78

�85

� 2.4x

�4.6x

1.3y

0.5y

14.63

�11.51

7.

, ,

Solution: ��1, 3, 2�

z ��425 �1

2

�2

�5

10

1�55

�110

55� 2y �

�425 �5

10

1

1

3

6�55

�165

55� 3x �

��5

10

1

�1

2

�2

1

3

6�55

��55

55� �1

D � �425 �12

�2

136� � 55�

4x

2x

5x

y

2y

2y

z

3z

6z

�5

10

1

,

Section 8.5 Applications of Matrices and Determinants 779

8.

Solution: �5, 8, �2�

z ��428 �2

2

�5

�2

16

4��82

�164

�82� �2

y ��428 �2

16

4

3

5

�2��82

��656

�82� 8

x ���2

16

4

�2

2

�5

3

5

�2��82

��401

�82� 5

D � �428 �2

2

�5

3

5

�2� � �82

�4x

2x

8x

2y

2y

5y

3z

5z

2z

�2

16

4

9.

Solution: ��2, 1, �1�

z � � 1�2

3

21

�3

�36

�11�10

��1010

� �1

y � � 1�2

3

�36

�11

3�1

2�10

�1010

� 1

x � � �36

�11

21

�3

3�1

2�10

��2010

� �2

D � � 1�2

3

21

�3

3�1

2� � 10�x

�2x

3x

2y

y

3y

3z

z

2z

�3

6

�11

,

10.

Solution: �0, 3, �2�

z �� 5

�1

3

�4

2

1

�14

10

1�33

� �66

33� �2

y �� 5

�1

3

�14

10

1

1

�2

1�33

�99

33� 3

x ���14

10

1

�4

2

1

1

�2

1�33

�0

33� 0

D � � 5

�1

3

�4

2

1

1

�2

1� � 33

�5x

�x

3x

4y

2y

y

z

2z

z

�14

10

1

11.

Solution: 0, �1

2,

1

2

z ��335 3

5

9

1

2

4�4

�1

2

y ��335 1

2

4

5

9

17�4

� �1

2

x ��124 3

5

9

5

9

17�4

� 0

D � �335 3

5

9

5

9

17� � 4�3x

3x

5x

3y

5y

9y

5z

9z

17z

1

2

4

,

12.

, ,

Solution: ��3, �1, 2�

z � � 12

�1

2�2

3

�7�8

8��18

� 2 y � � 12

�1

�7�8

8

�1�2

4��18

� �1 x � ��7�8

8

2�2

3

�1�2

4��18

� �3

D � � 1

2

�1

2

�2

3

�1

�2

4� � �18�x

2x

�x

2y

2y

3y

z

2z

4z

�7

�8

8

,

780 Chapter 8 Matrices and Determinants

13.

Solution: �1, 2, 1�

x � �606

122

2�3�1�

�18� 1, y � � 2

�13

606

2�3�1�

�18� 2, z � � 2

�13

122

606�

�18� 1

D � � 2�1

3

122

2�3�1� � �18�

2x

�x

3x

y

2y

2y

2z

3z

z

6

0

6

15. Vertices:

Area �1

2�031 0

1

5

1

1

1� �1

2�31 1

5� � 7 square units

�0, 0�, �3, 1�, �1, 5�

17. Vertices:

Area �1

2��2

2

0

�3

�3

4

1

1

1� �1

2�2��3

4

1

1� � 2��3

4

1

1� �1

2�14 � 14� � 14 square units

��2, �3�, �2, �3�, �0, 4�

14.

Cramer’s Rule does not apply.

D � �235 3

5

9

5

9

17� � 0

�2x

3x

5x

3y

5y

9y

5z

9z

17z

4

7

13

16. Vertices:

Area � �1

2�045 0

5

�2

1

1

1� � �1

2�45 5

�2� �33

2 square units

�0, 0�, �4, 5�, �5, �2�

18. Vertices:

Area � �1

2��2

1

3

1

6

�1

1

1

1� � �1

2�2� 6

�1

1

1� � �13 1

1� � �13 6

�1� � �1

2��14 � 2 � 19� �

31

2 square units

��2, 1�, �1, 6�, �3, �1�

19. Vertices:

Area �1

2�0524 12

03

111� �

1

2�1

2� 52

4

1

1� � 1� 52

4

0

3� �1

23

4�

15

2 �33

8 square units

0, 1

2, 5

2, 0, �4, 3�

20. Vertices:

Area � �1

2��4

6

6

�5

10

�1

1

1

1� � �1

26��5

10

1

1� � ��1���4

6

1

1� � ��4

6

�5

10� � 55 square units

��4, �5�, �6, 10�, �6, �1�

21. Vertices:

Area �1

2��2

2

�1

4

3

5

1

1

1� �1

2�� 2

�1

3

5� � ��2

�1

4

5� � ��2

2

4

3�� �1

2�13 � 6 � 14� �

5

2 square units

��2, 4�, �2, 3�, ��1, 5�

Section 8.5 Applications of Matrices and Determinants 781

22. Vertices:

Area � �1

2� 0

�1

3

�2

4

5

1

1

1� � �1

22��1

3

1

1� � ��1

3

4

5� � �1

2��8 � 17� �

25

2 square units

�0, �2�, ��1, 4�, �3, 5�

24. Vertices:

Area � �1

2��2

1

3

4

5

�2

1

1

1� � �1

2�2� 5

�2

1

1� � � 4

�2

1

1� � 3�45 1

1� � �1

2��14 � 6 � 3� �

23

2 square units

��2, 4�, �1, 5�, �3, �2�

23. Vertices:

Area � �1

2��3

2

3

5

6

�5

1

1

1� � �1

2��23 6

�5� � ��3

3

5

�5� � ��3

2

5

6�� � �1

2��28 � 0 � 28� � 28 square units

��3, 5�, �2, 6�, �3, �5�

25.

y �16

5 or y � 0

y �8 ± 8

5

±8 � 5y � 8

±8 � �5�2 � y� � 2��1�

±8 � �5�2y 1

1� � 2�12 1

1� 4 � ±

1

2��5

0

�2

1

2

y

1

1

1� 26.

y � 19 or y � 3

y � 11 ± 8

±8 � y � 11

±8 � �3y � 5 � 4y � 2 � 20 � 6

±8 � �3y � 5 � ��4y � 2� � 20 � 6

±8 � ��3

�1

5

y� � ��4

�1

2

y� � ��4

�3

2

5� 4 � ±

1

2��4

�3

�1

2

5

y

1

1

1�

27.

y � �3 or y � �11

y ��21 ± 12

3� �7 ± 4

±12 � 3y � 21

±12 � � y � 8� � ��2y � 24� � 5

±12 � � 1�8

�1y� � ��2

�8�3

y� � ��21

�3�1�

6 � ±12��2

1�8

�3�1

y

111� 28.

y � 6 or y � 0

y �12 ± 12

4� 3 ± 3

±12 � 4y � 12

±12 � �3 � y � 5y � 9

±12 � ��3

y

1

1� � � 5

�3

�3

y� 6 � ±

1

2� 1

5

�3

0

�3

y

1

1

1�

29. Vertices:

Area �1

2� 0

10

28

25

0

5

1

1

1� � 250 square miles

�0, 25�, �10, 0�, �28, 5� 30. Vertices:

Area � �1

2� 0

85

20

30

0

�50

1

1

1� � 3100 square feet

�0, 30�, �85, 0�, �20, �50�

782 Chapter 8 Matrices and Determinants

31. Points:

The points are collinear.

� 3

0

12

�1

�3

5

1

1

1� � 3��3

5

1

1� � 12��1

�3

1

1� � 3��8� � 12�2� � 0

�3, �1�, �0, �3�, �12, 5�

32. Points:

The points are not collinear.

��3

6

10

�5

1

2

1

1

1� � � 6

10

1

2� � ��3

10

�5

2� � ��3

6

�5

1� � 2 � 44 � 27 � �15 � 0

��3, �5�, �6, 1�, �10, 2�

33. Points:

The points are not collinear.

� 2

�4

6

�12

4

�3

1

1

1� � ��4

6

4

�3� � �26 �12

�3� � � 2

�4

�12

4� � �12 � 3 � 6 � �3 � 0

�2, �12�, ��4, 4�, �6, �3�

34. Points:

The points are collinear.

� 0

4

�2

1

�252

1

1

1� � �� 4

�2

1

1� � � 4

�2

�252� � �6 � 6 � 0

�0, 1�, �4, �2�, ��2, 52�

35. Points:

The points are collinear.

� 0

1

�1

2

2.4

1.6

1

1

1� � �2� 1

�1

1

1� � � 1

�1

2.4

1.6� � �2�2� � 4 � 0

�0, 2�, �1, 2.4�, ��1, 1.6�

36. Points:

The points are not collinear.

� 2

3

�1

3

3.5

2

1

1

1� � � 3

�1

3.5

2� � � 2

�1

3

2� � �23 3

3.5� � 9.5 � 7 � ��2� �1

2� 0

�2, 3�, �3, 3.5�, ��1, 2�

37.

y � �3

�3y � 9 � 0

2� y � 2� � 5��1� � ��8 � 5y� � 0

2� y

�2

1

1� � 5�45 1

1� � �45 y

�2� � 0

�245 �5

y

�2

1

1

1� � 0 38.

y � 3

�3y � �9

�25 � 3y � 24 � 6y � 10 � 0

��5

�3

y

5� � ��6

�3

2

5� � ��6

�5

2

y� � 0

��6

�5

�3

2

y

5

1

1

1� � 0

Section 8.5 Applications of Matrices and Determinants 783

39. Points:

Equation: �x

0

5

y

0

3

1

1

1� � ��x

5

y

3� � 5y � 3x � 0 ⇒ 3x � 5y � 0

�0, 0�, �5, 3�

41. Points:

Equation: � x

�4

2

y

3

1

1

1

1� � x�31 1

1� � y��4

2

1

1� � ��4

2

3

1� � 2x � 6y � 10 � 0 ⇒ x � 3y � 5 � 0

��4, 3�, �2, 1�

43. Points:

Equation: � x�

1252

y3

1

11

1� � x�31 1

1� � y��1252

1

1� � ��1252

3

1� � 2x � 3y � 8 � 0

��12, 3�, �5

2, 1�

45. The uncoded row matrices are the rows of the matrix on the left.

Solution: �52 10 27 �49 3 34 �49 13 27 �94 22 54 1 1 �7 0 �12 9 �121 41 55

TUENIRI

R B

V

T

OLIRECY

�20215

149

189

18200

220

20

15129

1853

25

� �11

�6

�102

0�1

3� � ��52�49�49�94

10

�121

103

13221

�1241

27342754

�79

55

�7 � 3

40. Points:

Equation: � x

0

�2

y

0

2

1

1

1� � �� x

�2

y

2� � ��2x � 2y� � 0 or x � y � 0

�0, 0�, ��2, 2�

42. Points:

Equation:

� x

10

�2

y

7

�7

1

1

1� � � 10

�2

7

�7� � � x

�2

y

�7� � � x

10

y

7� � �70 � 14 � ��7x � 2y� � 7x � 10y � 0 or 7x � 6y � 28 � 0

�10, 7�, ��2, �7�

44. Points:

Equation: �x23

6

y

4

12

1

1

1� � � 23

6

4

12� � �x

6

y

12� � �x23 y

4� � �16 � �12x � 6y� � 4x �23 y � 0 or 3x � 2y � 6 � 0

�23, 4�, �6, 12�

784 Chapter 8 Matrices and Determinants

46.

Solution: Uncoded matrices:Encoded matrices:

Encoded message:44 16 10 49 9 12 �55 �65 �2043 6 9 �38 �45 �13 �42 �47 �14

�44 16 10�, �49 9 12�, ��55 �65 �20��43 6 9�, ��38 �45 �13�, ��42 �47 �14�,1 � 3

�14 4 0�, �13 15 14�, �5 25 0��16 12 5�, �1 19 5�, �0 19 5�,1 � 3

�5 25 0��4

�3

3

2

�3

2

1

�1

1� � ��55 �65 �20�

�13 15 14��4

�33

2�3

2

1�1

1� � �49 9 12�

�14 4 0��4

�3

3

2

�3

1

1

�1

1� � �44 16 10�

�0 19 5��4

�3

3

2

�3

3

1

�1

1� � ��42 �47 �14�

�1 19 5��4

�3

3

2

�3

2

1

�1

1� � ��38 �45 �13�

�16 12 5��4

�3

3

2

�3

2

1

�1

1� � �43 6 9�

In Exercises 47–50, use the matrix A � [ 13

�1

27

�4

29

�7].

47.

Cryptogram: �6 �35 �69 11 20 17 6 �16 �58 46 79 67

�15 15 14� A � �46 79 67�

�20 0 14� A � �6 �16 �58�

�12 0 1� A � �11 20 17�

�3 1 12� A � ��6 �35 �69�

C[3

A1

L12]

L

[12__0

A1]

T

[20__0

N14]

O

[15O15

N14]

48.

—CONTINUED—

�7 0 4��13

�1

27

�4

29

�7� � �3 �2 �14�

�2 5 18��13

�1

27

�4

29

�7� � ��1 �33 �77�

�9 3 5��13

�1

27

�4

29

�7� � �13 19 10�

�9 3 5� �2 5 18� �7 0 4� �5 1 4� �0 1 8� �5 1 4�

I C E B E R G __ D E A D __ A H E A D

Section 8.5 Applications of Matrices and Determinants 785

48. —CONTINUED—

Cryptogram:4 1 �9 �5 �25 �47 4 1 �913 19 10 �1 �33 �77 3 �2 �14

�5 1 4��13

�1

27

�4

29

�7� � �4 1 �9�

�0 1 8��13

�1

27

�4

29

�7� � ��5 �25 �47�

�5 1 4��13

�1

27

�4

29

�7� � �4 1 �9�

49.

Cryptogram: �5 �41 �87 91 207 257 11 �5 �41 40 80 84 76 177 227

� 11 25 10� A � � 76 177 227�

� 20 28 14� A � � 40 80 84�

� 12 29 18� A � � 11 �5 2�41�

� 16 25 10� A � � 91 207 257�

� 18 21 16� A � ���5 �41 �87�

H[8

A1

P16]

P

[16Y25

__0]

B[2

I9

R18]

T

[20H8

D4]

A[1

Y25

__0]

50.

Cryptogram: 58 122 139 1 �37 �95 40 67 55 23 17 �19 47 88 88 14 21 11

�15 1 4��1

3

�1

2

7

�4

2

9

�7� � �14 21 11�

�5 18 12��1

3

�1

2

7

�4

2

9

�7� � �47 88 88�

�0 15 22��1

3

�1

2

7

�4

2

9

�7� � �23 17 �19�

�9 15 14��1

3

�1

2

7

�4

2

9

�7� � �40 67 55�

�18 1 20��1

3

�1

2

7

�4

2

9

�7� � �1 �37 �95�

�15 16 5��1

3

�1

2

7

�4

2

9

�7� � �58 122 139�

�15 16 5� �18 1 20� �9 15 14� �0 15 22� �5 18 12� �15 1 4�

O P E R A T I O N _ O V E R L O A D

786 Chapter 8 Matrices and Determinants

51.

Message: HAPPY NEW YEAR�11642529234055

211125053467592

� ��53

2�1� � �

816251423251

1160505

18

HPYNWYA

AP__E__ER

A�1 � �1

3

2

5��1

� ��5

3

2

�1�

52.

Message: BRONCOS WIN SUPER BOWL

��199 82���3�7

25� � �23 12� W L

��115 49���3

�7

2

5� � �2 15� B O

��90 36���3

�7

2

5� � �18 0� R __

��115 47���3�7

25� � �16 5� P E

��242 101���3�7

25� � �19 21� S U

��70 28���3

�7

2

5� � �14 0� N __

��178 73���3

�7

2

5� � �23 9� W I

��95 38���3�7

25� � �19 0� S __

��120 51���3�7

25� � �3 15� C O

��173 72���3

�7

2

5� � �15 14� O N

��136 58���3

�7

2

5� � �2 18� B R

A�1 � ��3

�7

2

5�

53.

Message: CLASS IS CANCELED�9

3828

�80�64

9

�1�19�92521

�5

�9�19�19

4131

�4� �

�2�3�2

�3�3�4

�1�1�1� � �

3199335

121919154

100

14120�

CSICCE

L S S A E D

A____NL__

A�1 � �11

�6

�102

0�1

3��1

� ��2�3�2

�3�3�4

�1�1�1�

Section 8.5 Applications of Matrices and Determinants 787

54.

Message: HAVE A GREAT WEEKEND

�42 �48 32��11

4

�8

2

1

�1

�8

�3

6� � �14 4 0� N D __

�35 �23 36��11

4

�8

2

1

�1

�8

�3

6� � �5 11 5� E K E

�20 21 38��11

4

�8

2

1

�1

�8

�3

6� � �0 23 5� __ W E

�95 �118 71��11

4

�8

2

1

�1

�8

�3

6� � �5 1 20� E A T

�72 �76 61��11

4

�8

2

1

�1

�8

�3

6� � �0 7 18� __ G R

�19 �25 13��11

4

�8

2

1

�1

�8

�3

6� � �5 0 1� E __ A

�112 �140 83��11

4

�8

2

1

�1

�8

�3

6� � �8 1 22� H A V

A�1 � �11

4

�8

2

1

�1

�8

�3

6�

55.

Message: SEND PLANES�20

�12

1

62

17

�56

�25

143

�15

�104

�65

181� �

�13

12

�5

6

�5

2

4

�3

1� � �

19

4

12

5

5

0

1

19

14

16

14

0�

S

D

L

E

E

__

A

S

N

P

N

__

A�1 � �1

3

�1

2

7

�4

2

9

�7�

�1

� ��13

12

�5

6

�5

2

4

�3

1�

56.

Message: RETURN AT DAWN

�65 144 172���13

12

�5

6

�5

2

4

�3

1� � �23 14 0� W N __

�11 24 29���13

12

�5

6

�5

2

4

�3

1� � �0 4 1� __ D A

��17 �73 �131���13

12

�5

6

�5

2

4

�3

1� � �0 1 20� __ A T

�61 112 106���13

12

�5

6

�5

2

4

�3

1� � �21 18 14� U R N

�13 �9 �59���13

12

�5

6

�5

2

4

�3

1� � �18 5 20� R E T

788 Chapter 8 Matrices and Determinants

58. (a)

System:

(b)

The least squares regression parabola is

(c)

(d) The intersection of the regression parabola and the line is about so the number of cases waiting to be tried will reach 10,000 in about 2004.

t � 4.3,y � 10,000

08,000

8

12,000

y � 9.5t2 � 201.5t � 8965.

a � �335 359

27,54727,98846,800�

4�

384

� 9.5

b � �3 3 5

27,54727,988

46,800

59

17�4

�806

4� 201.5

c � �27,54727,98846,800

359

59

17�4

�35,860

4� 8965

D � �335 359

59

17� � 4

�3c � 3b � 5a � 27,5473c � 5b � 9a � 27,9885c � 9b � 17a � 46,800

�n

i�1xi

2yi � 02�8965� � 12�9176� � 22�9406� � 46,800

�n

i�1xi yi � 0�8965� � 1�9176� � 2�9406� � 27,988

�n

i�1xi

4 � 04 � 14 � 24 � 17; �n

i�1yi � 8965 � 9176 � 9406 � 27,547

n � 3; �n

i�1xi � 0 � 1 � 2 � 3; �

n

i�1xi

2 � 02 � 12 � 22 � 5; �n

i�1xi

3 � 03 � 13 � 23 � 9;

57. Let A be the matrix needed to decode the message.

Message: MEET ME TONIGHT RON�

8

�15

�13

5

5

5

�1

20

�18

1

21

�10

�13

10

25

19

6

40

�18

16

� ��1

1

�2

1� � �

13

5

0

5

20

14

7

20

0

15

5

20

13

0

15

9

8

0

18

14

M

E

__

E

T

N

G

T

__

O

E

T

M

__

O

I

H

__

R

N

A � ��18

1

�18

16��1

� 0

15

18

14� � �� 81351

270

� 115115

� � 0

15

18

14� � ��1

1

�2

1�

��18

1

�18

16� A � � 0

15

18

14�

O

R

N

2 � 2

Section 8.5 Applications of Matrices and Determinants 789

59. False. In Cramer’s Rule, the denominator is the determinant of the coefficient matrix.

61. False. If the determinant of the coefficient matrix is zero, the system has either no solution or infinitely many solutions.

63.

x

Solution: ��6, 4�

�6�

�5x � 35y

5x � y

�34y

y

�x � 7�4�

�110

�26

�136

4

�22

�5�Eq.1

Add equations.

Equation 1

Equation 2��x

5x

7y

y

�22

�26

60. True. If the determinant of the coefficient matrix is zero,the solution of the system would result in division by zerowhich is undefined.

62. Answers will vary. To solve a system of linear equationsyou can use graphing, substitution, elimination, elementaryrow operations on an augmented matrix (Gaussian elimination with back–substitution or Gauss-Jordan elimination), the inverse of a matrix, or Cramer’s Rule.

64.

Solution: �5, �12�

3�5� � 8y � 11 ⇒ 8y � �4 ⇒ y � �12

�3�Eq.1��2�Eq.2Add equations.

� 9x

4x

13x

24y

24y

x

33

32

656513 � 5

Equation 1Equation 2� 3x

�2x

8y

12y

11

�16

65.

Solution: ��1, 0, �3�

�xyz� � A�1�

�14�1

�11� � ��1

0�3�

�1

72��1

13

22

9

27

18

7

�19

�10�

A�1 � ��1

4

5

�3

2

�3

5

�1

2�

�1

��x

4x

5x

3y

2y

3y

5z

z

2z

�14

�1

�11

66.

Solution: �2, �2, 5�

�xyz� � A�1�

7�5

�37� � �2

�25�

A�1 � �5

�24

�13

10

�11

�5��1

� �25872

293287

5297

291829

�2

871

29

�1387

��

5x

�2x

4x

y

3y

10y

z

z

5z

7

�5

�37

67. Objective function:

Constraints:

6x � y ≤ 40

x � 6y ≤ 30

y ≥ 0

x(0, 0) , 0 ( (20

3

(0, 5)(6, 4)

2

4

6

2 4 6

y x ≥ 0

z � 6x � 4y

The minimum value of 0 occurs at

The maximum value of 52 occurs at �6, 4�.

�0, 0�.

At �203 , 0�: z � 6�20

3 � � 4�0� � 40

At �6, 4�: z � 6�6� � 4�4� � 52

At �0, 5�: z � 6�0� � 4�5� � 20

At �0, 0�: z � 6�0� � 4�0� � 0

68. Objective function:

Constraints:

Since the region is unbounded,there is no maximum value of the objective function. To find the minimum value, check the vertices.

The minimum value of 46 occurs at �3, 4�.

At �15, 0� : z � 6(15� � 7�0� � 90

At �3, 4� : z � 6�3� � 7�4� � 46

At �0, 8� : z � 6�0� � 7�8� � 56

x � 3y ≥ 15

4x � 3y ≥ 24

y ≥ 0

x

4

8

12

16

4 8 12

(15, 0)

(3, 4)

(0, 8)

y x ≥ 0

z � 6x � 7y

Review Exercises for Chapter 8

790 Chapter 8 Matrices and Determinants

1.

Order: 3 � 1

��4

05� 2.

Since the matrix has two rows andfour columns, its order is 2 � 4.

� 3

�2

�1

7

0

1

6

4� 3.

Order: 1 � 1

�3�

4.

Since the matrix has one row andfive columns, its order is 1 � 5.

�6 2 �5 8 0� 5.

�3

5

�10

4

..

.

..

.15

22�

�3x

5x

10y

4y

15

22

6.

�8

3

5

�7

�5

3

4

2

�3

���

12

20

26�

�8x

3x

5x

7y

5y

3y

4z

2z

3z

12

20

26

7.

�5x

4x

9x

y

2y

4y

7z

2z

�9

10

3

�5

4

9

1

2

4

7

0

2

���

�9

10

3� 8.

�13x

x

4x

16y

21y

10y

7z

8z

4z

3w

5w

3w

2

12

�1

�13

1

4

16

21

10

7

8

�4

3

5

3

���

2

12

�1�

9.

�12R3 →

�100

210

311�

2R2 � R3 →

�100

210

31

�2�

�2R1 � R3 → �

100

21

�2

31

�4�

R1

R2 �102

212

312�

�0

1

2

1

2

2

1

3

2� 10.

�17R2 →�

100

210

4107

0�

�R2 � R3→�

100

2�7

0

4�10

0�

�3R1 � R2→�R1 � R3→�

100

2�7�7

4�10�10�

14R1 →

�12R3 →�

131

2�1�5

42

�6��

43

�2

8�110

162

12�

11.

Solution: �5, 2, 0�

x � 2�2� � 3�0� � 9 ⇒ x � 5

y � 2�0� � 2 ⇒ y � 2

�100

210

3�2

1

���

920�

⇒⇒⇒

�x � 2y � 3z � 9y � 2z � 2

z � 012.

Solution: ��38, 8, �2�

x � �38

x � 3�8� � 9��2� � 4

y � 8

y � ��2� � 10

�x � 3yy

9zzz

410

�2

Review Exercises for Chapter 8 791

13.

Solution: ��40, �5, 4�

x � 5��5� � 4�4� � 1 ⇒ x � �40

y � 2�4� � 3 ⇒ y � �5

�100

�510

421

���

134�

⇒⇒⇒

�x � 5y � 4z � 1y � 2z � 3

z � 414.

Solution: ��50, �6, 1�

x � �50

x � 8��6� � �2

y � �6

y � 1 � �7

�x � 8y y � z

z

�2�7

1

15.

Solution: �10, �12�

x � 8��12� � �86 ⇒ x � 10

y � �12

�x � 8y � �86y � �12

19R2 →�1

081

��

�86�12�

R1 � R2 → �10

89

��

�86�108�

4R2 � R1 →

�1

�181

��

�86�22�

� 5�1

41

��

2�22� 16.

Solution: ��9, �4�

x �52��4� � 1 ⇒ x � �9

y � �4

�x �52y

y

1

�4

2R3 → �1

0

�52

1

1

�4�

�3R1 � R2 → �1

0

�52

12

1

�2�

12 R1 →

�1

3

�52

�7

1

1� �2

3

�5

�7��

2

1�

17.

Solution: ��0.2, 0.7� � ��15, 7

10� x � 2�0.7� � 1.2 ⇒ x � �0.2

y � 0.7

�x � 2y

y

1.2

0.7

�17R2 →�1

021

��

1.20.7�

�2R1 � R2 → �10

2�7

��

1.2�4.9�

�R2 � R1 → �1

22

�3��

1.2�2.5�

10R1 →10R2 → �3

2�1�3

��

�1.3�2.5�

�0.30.2

�0.1�0.3

��

�0.13�0.25� 18.

Solution: �0.6, 0.5� � �35, 12�

y

x � 0.5�0.5��

0.5

0.35 ⇒ x � 0.6

�x � 0.5yy

0.350.5

�1

0.3 R2 → �1

0�0.5

1��

0.350.5�

5R1 →

�2R1 � R2 → �1

0�0.5�0.3

��

0.35�0.15�

�0.2

0.4

�0.1

�0.5��

0.07

�0.01�

792 Chapter 8 Matrices and Determinants

19.

Solution: �5, 2, �6�

x �32�2� �

12��6� � 5 ⇒ x � 5

y �23��6� � �2 ⇒ y � 2

z � �6

319R3 →

�1

00

32

10

1223

1

��

5

�2�6

8R2 � R3 →�1

0

0

32

1

0

1223

193

5

�2

�38�

12R1 →

�16R2 →

�1

00

32

1�8

1223

1

��

5

�2�22

�R1 � R2 →�2R1 � R3 → �

200

3�6�8

1�4

1

���

1012

�22�

�224

3�3�2

1�3

3

���

1022

�2�

21.

Let then:

Solution: where a is any real number��2a �32, 2a � 1, a�

x � 2a �32 ⇒ x � �2a �

32

y � 2a � 1 ⇒ y � 2a � 1

z � a,

12R1 → �1

00

010

2�2

0

���

32

10�

�R2 � R1 →

2R2 � R3 → �

2

0

0

0

1

0

4

�2

0

���

3

1

0�

�R1 � R2 →�R1 � R3 →

�2

0

0

1

1

�2

2

�2

4

���

4

1

�2�

�2

2

2

1

2

�1

2

0

6

���

4

5

2�

20.

Solution: �12, � 1

3, 1� x � 3�1� �

72 ⇒ x �

12

y � 1 � � 43 ⇒ y � � 1

3

z � 1

�x

y

3z

z

z

72

�43

1

12R1 →

�13R2 →

�128R3 →

�1

0

0

0

1

0

3

�1

1

72

�43

1�

R2 � R1 →

�3R2 � R3 → �

2

0

0

0

�3

0

6

3

�28

���

7

4

�28�

�3R1 � R2 →�6R1 � R3 →

�2

0

0

3

�3

�9

3

3

�19

���

3

4

�16�

�2

6

12

3

6

9

3

12

�1

���

3

13

2�

22.

Because the last row consists of all zeros except for the last entry, the system is inconsistent and there is no solution.

5R2 � R3 → �

100

210

630

���

121�

�2R1 � R2 →�3R1 � R3 →

�100

21

�5

63

�15

���

12

�9�

�123

251

6153

���

14

�6�

Review Exercises for Chapter 8 793

23.

Solution: �1, 0, 4, 3�

x � 0 � 3�3� � �8 ⇒ x � 1

y � 19�3� � �57 ⇒ y � 0

z � 13�3� � �35 ⇒ z � 4

w � 3

119R4 →

�1000

1100

0010

�3�19�13

1

����

�8�57�35

3�

2R3�R4 →

�1000

1100

0010

�3�19�13�19

����

�8�57�35�57

R4

R3

�1000

1100

001

�2

�3�19�13

7

����

�8�57�35

13�

R2 � R4 →�1000

1100

00

�21

�3�19

7�13

����

�8�57

13�35

�3R4 � R2 →

�1000

110

�1

00

�21

�3�19

76

����

�8�57

1322

� �3R1 � R3 →

�R1 � R4 →�1000

1�2

0�1

03

�21

�3�1

76

����

�89

1322

�R4 � R1

�1031

1�2

30

03

�21

�3�1�2

3

����

�89

�1114

� �2031

1�2

30

13

�21

0�1�2

3

����

69

�1114

� 24.

Because the last row consists of all zeros except for thelast entry, the system is inconsistent and there is no solution.

�R3 � R4 →

�1

0

0

0

2

1

�4

0

0

�1

1

0

1

0

�2

0

����

3

0

�12

9�

13R2 →

�4R1 � R3 →�2R1 � R4 →

�1

0

0

0

2

1

�4

�4

0

�1

1

1

1

0

�2

�2

����

3

0

�12

�3�

�1

0

4

2

2

�3

4

0

0

3

1

1

1

0

2

0

����

3

0

0

3�

25.

15R2 →

�100

�119

�21

12

���

�109�

�2R1 � R2 →�5R1 � R3 → �

1

0

0

�1

5

9

�2

5

12

���

�1

0

9�

�R1 →

125

�134

�212

���

�1�2

4�

��1

2

5

1

3

4

2

1

2

���

1

�2

4�

Solution: �2, �3, 3�

x � 2, y � �3, z � 3

R3 � R1 →�R3 � R2 → �

1

0

0

0

1

0

0

0

1

���

2

�3

3�

13R3 →�

100

010

�111

���

�103�

R2 � R1 →

�9R2 � R3 → �1

0

0

0

1

0

�1

1

3

���

�1

0

9�

794 Chapter 8 Matrices and Determinants

26.

Solution: �3142, 5

14, 1384�

x

y

z

3142

514

1384

R3 � R1 →

�2R3 � R2 → �1

0

0

0

1

0

0

0

1

3142

514

1384�

17R3 →

�1

0

0

0

1

0

�1

2

1

712

23

1384�

�R2 � R1 →

2R2 � R3 →

�1

0

0

0

1

0

�1

2

7

712

23

1312�

�16R2 →

�1

0

0

1

1

�2

1

2

3

54

23

�14�

�4R1 � R2 →

�5R1 � R3 →

�1

0

0

1

�6

�2

1

�12

3

54

�4

�14�

14R1 →

�145

1�2

3

1�8

8

���

54

16�

�445

4�2

3

4�8

8

���

516�

�4x

4x

5x

4y

2y

3y

4z

8z

8z

5

1

6

27.

Solution: �2, 3, �1�

z � �1y � 3,x � 2,

�R3 � R1 →3R3 � R2 → �

100

010

001

���

23

�1�

4R2 � R1 →

�100

010

1�3

1

���

16

�1�

�14R3 →�

100

�410

13�3

1

���

�236

�1�

7R2 � R3 → �100

�410

13�3�4

���

�2364�

122R2 →�

100

�41

�7

13�317

���

�236

�38�

R3

R2

�100

�422

�7

13�66

17

���

�23132

�38�

R1 � R2 →�5R1 � R3 → �

100

�4�722

1317

�66

���

�23�38132�

R2 � R1 →

1�1

5

�4�3

2

134

�1

���

�23�15

17�

�2

�15

�1�3

2

94

�1

���

�8�15

17�

28.

�5R1 � R2 →3R1 � R3 →

�100

�13

�2

�419

�5

���

8�6

4�

�1R1 →

15

�3

�1�2

1

�4�1

7

���

834

�20�

R3

R1

��1

5�3

1�2

1

4�1

7

���

�834

�20�

��3

5�1

1�2

1

7�1

4

���

�2034

�8��

�3x

5x

�x

y

2y

y

7z

z

4z

�20

34

�8

Solution: �6, �2, 0�

x � 6, y � �2, z � 0

73R3 � R1 →

�193 R3 � R2 →

�1

0

0

0

1

0

0

0

1

���

6

�2

0�

3

23R3 →

�1

0

0

0

1

0

73

193

1

6

�2

0�

R2 � R1 →

2R2 � R3 →

�1

0

0

0

1

0

73

193

233

6

�2

0�

13R2 →

1

0

0

�1

1

�2

�4193

�5

���

8

�2

4�

Review Exercises for Chapter 8 795

29. Use the reduced row-echelon form feature of a graphing utility.

Solution: �2, 6, �10, �3�

w � �3z � �10,y � 6,x � 2,

�3150

�16

�14

541

�1

�2�1

3�8

����

�441

�1558

� ⇒ �1000

0100

0010

0001

����

26

�10�3

31. ��1y

x9� � ��1

�7129� ⇒ x � 12 and y � �7

33.

x � 3 �

4y �

y � 5 �

6x �

5x � 144166

� x � 1 and y � 11

�x � 3

0�2

�4�3

y � 5

4y2

6x� � �

5x � 10

�2

�4�316

4426�

35. (a)

(b)

(c)

(d) A � 3B � �23

�25� � 3��3

12108� � �2

3�2

5� � ��936

3024� � ��7

392829�

4A � 4�23

�25� � � 8

12�820�

A � B � �23

�25� � ��3

12108� � � 5

�9�12�3�

A � B � �23

�25� � ��3

12108� � ��1

158

13�

30. Use the reduced row-echelon form feature of the graphing utility.

The system is inconsistent and there is no solution.

�4

1

1

�2

12

6

6

�10

2

4

1

�2

����

20

12

8

�10� ⇒ �

1

0

0

0

0

1

0

0

0

0

1

0

����

0

0

0

1�

32. ��1

x

�4

0

5

y� � �

�1

8

�4

0

5

0� ⇒ x � 8, y � 0

34.

6

2

�4

12x

x � 10

2y� x � 12, y � �2

��9

0

6

4

�3

�1

2

7

1

�5

�4

0� � �

�9

012x

4

�3

�1

x � 10

7

1

�5

2y

0�

36. (a)

(b)

(c)

(d) � �175356

40

122 92�� �

5�711

422� � �

126045

3612090� A � 3B � �

5�711

422� � 3�

42015

124030�

4A � 4�5

�711

422� � �

20�28

44

1688�

� �1

�27�4

�8

�38 �28�� �

5 � 4�7 � 2011 � 15

4 � 122 � 40

2 � 30� A � B � �5

�711

422� � �

42015

124030�

� �9

1326

1642 32�� �

5 � 4�7 � 2011 � 15

4 � 122 � 40

2 � 30� A � B � �5

�711

422� � �

42015

124030�

796 Chapter 8 Matrices and Determinants

37. (a)

(b)

(c)

(d) A � 3B � �5

�711

422� � 3 �

04

20

31240� � �

5�711

422� � �

01260

936

120� � �55

71

1338

122�

4A � 4�5

�711

422� � �

20�28

44

1688�

A � B � �5

�711

422� � �

04

20

31240� � �

5�11�9

1�10�38�

A � B � �5

�711

422� � �

04

20

31240� � �

5�331

71442�

38. (a) is not possible. A and B do not have the same order.

(b) is not possible. A and B do not have the same order.

(c)

(d) is not possible. A and B do not have the same order.A � 3B

4A � 4�6 �5 7� � �24 �20 28�

A � B

A � B

39. � 7�1

35� � �10

14�20�3� � � 7 � 10

�1 � 143 � 205 � 3� � �17

13�17

2�

40. Since the matrices are not of the same order, the operation cannot be performed.

41. �2�1

5

6

2

�4

0� � 8�

7

1

1

1

2

4� � �

�2

�10

�12

�4

8

0� � �

56

8

8

8

16

32� � �

54

�2

�4

4

24

32�

42.

� ��8 � 10

2 � 15

0 � 30

1 � 0

�4 � 5

6 � 60

�8 � 20

�12 � 5

0 � 40� � �

2

�13

�30

1

1

�54

12

�17

40�

� �8

�2

0

�1

4

�6

8

12

0� � 5�

�2

3

6

0

�1

12

�4

1

�8� � �

�8

2

0

1

�4

6

�8

�12

0� � �

10

�15

�30

0

5

�60

20

�5

40�

43. 3�8

1

�2

3

5

�1� � 6�4

2

�2

7

�3

6� � �24

3

�6

9

15

�3� � �24

12

�12

42

�18

36� � �48

15

�18

51

�3

33�

44.

�5

�2

7

8

0

�2

2� � 4�

4

6

�1

�2

11

3� � �

6

�11

�44

�8

54

2� 45.

� ��14

7

�17

�4

�17

�2�

X � 3A � 2B � 3��4

1

�3

0

�5

2� � 2�

1

�2

4

2

1

4�

Review Exercises for Chapter 8 797

46.

�16�

�13�2

0

6�17

20� � ��

136

�13

0

1

�176

103

� X �

16

�4A � 3B� �164�

�4

1

�3

0

�5

2� � 3�

1

�2

4

2

1

4� �

1

6��16

4

�12

0

�20

8� � �

3

�6

12

6

3

12� �

1

6��16 � 3

4 � 6

�12 � 12

0 � 6

�20 � 3

8 � 12�

47.

�1

3 �

9

�4

10

2

11

0� � �

3

�43

103

23

113

0�

X �1

3 �B � 2A� �

1

3 �

1

�2

4

2

1

4� � 2 �

�4

1

�3

0

�5

2�

48.

�13�

�1312

�26

�10�15�16� � �

�133

4

�263

�103

�5

�163�

X �13

�2A � 5B� �132�

�41

�3

0�5

2� � 5�1

�24

214� �

13�

�82

�6

0�10

4� � ��510

�20

�10�5

�20� �13�

�8 � 52 � 10

�6 � 20

0 � 10�10 � 5

4 � 20�

49. A and B are both so AB exists.

AB � �23

�25��

�312

108� � �2��3� � ��2��12�

3��3� � 5�12� 2�10� � ��2��8�

3�10� � 5�8�� � ��3051

470�

2 � 2

51. Since A is and is , AB exists.

AB � �5

�711

422�� 4

201240� � �

5�4� � 4�20��7�4� � 2�20�11�4� � 2�20�

5�12� � 4�40��7�12� � 2�40�11�12� � 2�40�� � �

1001284

220�4212�

2 � 2B3 � 2

50. Not possible because the number of columns of A does not equal the number of rows of B.

52. AB � �6 �5 7���1

48� � �6��1� � 5�4� � 7�8�� � �30�

53.

� �14

14

36

�2

�10

�12

8

40

48�

�1

5

6

2

�4

0� �6

4

�2

0

8

0� � �1�6� � 2�4�

5�6� � ��4��4�6�6� � �0��4�

1��2� � 2�0�5��2� � ��4��0�

6��2� � �0��0�

1�8� � 2�0�5�8� � ��4��0�

6�8� � �0��0��

54. Not possible because the number of columns of the first matrix does not equal the numberof rows of the second matrix.

798 Chapter 8 Matrices and Determinants

55.

� �44

20

4

8�

�1

2

5

�4

6

0� �6

�2

8

4

0

0� � �1�6� � 5��2� � 6�8�

2�6� � 4��2� � 0�8�1�4� � 5�0� � 6�0�2�4� � 4�0� � 0�0��

56.

� �4

0

0

6

6

0

3

�10

6�

�1

0

0

3

2

0

2

�4

3��

4

0

0

�3

3

0

2

�1

2� � �

1�4�0

0

1��3� � 3�3�2�3�

0

1�2� � 3��1� � 2�2�2��1� � ��4��2�

3�2��

57. �4

6� �6 �2� � �4�6�6�6�

4��2�6��2�� � �24

36

�8

�12�

58.

� �4 10�

�4 �2 6���2

0

2

1

�3

0� � �4��2� �2�0� � 6�2� 4�1� � 2��3� � 6�0��

59.

� � 1

12

17

36�

� �2�2� � 1��3�6�2� � 0

2�6� � 1�5�6�6� � 0�

�2

6

1

0� � 4

�3

2

1� � ��2

0

4

4� � �2

6

1

0� � 2

�3

6

5�

60.

� � �12

�246

9

144�

� � �3�15� � 3�11��12�15� � 6�11�

�3��9� � 3��6��12��9� � 6��6��

� � �3�12

3�6��

1511

�9�6�

�3�1

4

�1

2��0

1

3

2��1

5

0

�3� � � �3

�12

3

�6��0�1� � 3�5�1�1� � 2�5�

0�0� � 3��3�1�0� � 2��3��

61. �4

11

12

1

�7

3� �3

2

�5

�2

6

�2� � �14

19

42

�22

�41

�66

22

80

66� 62. ��2

4

3

�2

10

2� �1

�5

3

1

2

2� � �13

20

24

4�

63. � �7638

11495

13376� 0.95A � 0.95�80

40120100

14080� 64. � �

9660

108

843672

10896

120

482460�1.2A � 1.2�

805090

703060

9080

100

402050�

Review Exercises for Chapter 8 799

65.

The merchandise shipped to warehouse 1 is worth and the merchandise shipped to warehouse 2 is worth $303,150.

$274,150,

BA � �10.25 14.50 17.75��820065005400

740098004800� � �$274,150 $303,150�

66. (a)

(b)

Your cost with company A is $22.00. Your cost with company B is $22.80.

TC � �120 80 20��0.070.100.28

0.0950.080.25� � �22 22.8�

T � �120 80 20�

67.

� �10

01� � I

BA � ��2

7

�1

4���4

7

�1

2� � ��2��4� � ��1��7�7��4� � 4�7�

�2��1� � ��1��2�7��1� � 4�2��

� �10

01� � I

AB � ��4

7

�1

2���2

7

�1

4� � ��4��2� � ��1��7�7��2� � 2�7�

�4��1� � ��1��4�7��1� � 2�4��

68.

BA � � �2�11

15� � 5

11�1�2� � �1

001� � I

AB � � 511

�1�2� � �2

�1115� � �1

001� � I

69.

� �100

010

001� � I

� ��2�1� � ��3��1� � 1�6�

3�1� � 3�1� � ��1��6�2�1� � 4�1� � ��1��6�

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

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

BA � ��2

32

�334

1�1�1��

116

102

013�

� �100

010

001� � I

� �1��2� � 1�3� � 0�2�1��2� � 0�3� � 1�2�6��2� � 2�3� � 3�2�

1��3� � 1�3� � 0�4�1��3� � 0�3� � 1�4�6��3� � 2�3� � 3�4�

1�1� � 1��1� � 0��1�1�1� � 0��1� � 1��1�6�1� � 2��1� � 3��1��

AB � �1

1

6

1

0

2

0

1

3��

�2

3

2

�3

3

4

1

�1

�1�

800 Chapter 8 Matrices and Determinants

70.

BA � ��2

�3

2

1

1

�2

1212

�12

� �1

�18

�10

�4

0�1

2� � �100

010

001� � I

� �100

010

001� � IAB � �

1�1

8

�10

�4

0�1

2���2

�3

2

1

1

�2

1212

�12

71.

A�1 � �45

�5�6�

56R2 � R1 → �1

001

��

45

�5�6� � �I � A�1�

�6R2 → �1

0

�56

1

��

�16

5

0

�6�

5R1 � R2 → �1

0

�56

�16

�16

�56

0

1� �

16R1 → � 1

�5�

56

4��

�16

001�

�A � I� � ��6�5

54

��

10

01� 72.

A�1 � � 3�2

5�3�

�R2 � R1 → �1

001

��

3�2

5�3� � �I � A�1�

�2R1 � R2 → �1

011

��

1�2

2�3�

2R2 � R1 →

�12

13

��

10

21�

�A � I� � ��32

�53

��

10

01�

73.

A�1 � �13

�125

6�5

2

�43

�1�

�4R3 � R1 →

3R3 � R2 →�R3 →

�100

010

001

���

13�12

5

6�5

2

�43

�1� � �I � A�1�

�2R2 � R1 →

�2R2 � R3 → �

100

010

�43

�1

���

�73

�5

�21

�2

001�

�3R1 � R2 →�R1 � R3 →

�100

212

235

���

�131

010

001�

�R1 →

�131

274

297

���

�100

010

001�

�A � I� � ��1

31

�274

�297

���

100

010

001�

Review Exercises for Chapter 8 801

75. �2

�1

2

0

1

�2

3

1

1�

�1

� �12

12

0

�1

�23

23

�12

�56

13� 76.

does not exist.A�1

A � �1

2

�1

4

�3

18

6

1

16�

77.

� ��3

17

�1

6�2

�152.5

�5.52

14.5�2.5

3.5�1

�9.51.5

�143

�1

3442

121

�1

662

�2�

�1

� ��3

1

7

�1

6

�2

�1552

�112

2292

�52

72

�1

�19232

79.

A�1 �1

�7�2� � 2��8��2

8

�2

�7� �1

2�2

8

�2

�7� � �1

4

�1

�72�

A � ��7�8

22�

78.

A�1���2.5

�414.5�1

34.5

�161

711

�403

�2�312

�1�

A � �841

�1

0�2

24

2011

8�2

41�

80.

A�1 �1

10�3� � 4�7� �3

�7�410� �

12

� 3�7

�410� � �

32

�72

�2

5�

ad � bc � �10��3� � �4��7� � 2

A � �107

43�

74.

A�1 � �1

�1�1

11�7

�14

8�5

�10�

�R3 � R1 →

R3 � R2 → �100

010

001

���

1�1�1

11�7

�14

8�5

�10� � �I � A�1�

12R1 →

�R3 → �

100

010

1�1

1

���

00

�1

�37

�14

�25

�10�

�R2 � R1 →

2R2 � R3 → �200

010

2�1�1

���

001

�67

14

�45

10�

5R1 � 2R2 → �200

11

�2

1�1

1

���

001

170

150�

R2 � R1 →

�2

�50

1�2�2

1�3

1

���

001

110

100�

R3

R1

�7

�50

3�2�2

4�3

1

���

001

010

100�

�A � I� � �0

�57

�2�2

3

1�3

4

���

100

010

001�

802 Chapter 8 Matrices and Determinants

81.

� � 21

10

20316�

A�1 �1

�12��6� � 20� 3

10�� �6

�310

�20

�12� � �

1

3� �6

�3

10

�20

�12�

A � ��12

310

20

�6� 82.

A�1 �14

��8345

�52

�34� � ��2

315

�58

�3

16�

ad � bc � �34�

83 � 5

2�45 � 2 � 2 � 4

A � ��34

�45

52

�83�

83.

Solution: �36, 11�

� �7�8� � 4��5�2�8� � 1��5�� � �36

11�

�x

y� � ��1

2

4

�7��1� 8

�5� � �7

2

4

1��8

�5�

��x

2x

4y

7y

8

�5

85.

Solution: ��6, �1�

� ��17�8� � ��10���13��5�8� � ��3���13�� � ��6

�1�

�xy� � ��3

510

�17��1� 8

�13� � ��17�5

�10�3��

8�13�

��3x

5x

10y

17y

8

�13

84.

Solution: �2, �3�

�xy� � � 5

�9�1

2��1� 13

�24� � �29

15� � 13

�24� � � 2�3�

� 5x

�9x

y

2y

13

�24

86.

Solution: ��2, 1�

� � �92

�192

�1

�2� ��1047� � ��2

1�

�xy� � � 4

�19�2

9��1

��1047�

� 4x

�19x

2y

9y

�10

47

87.

Solution: �2, �1, �2�

� ��1�6� � 1��1� � 1�7�

3�6� �83��1� �

73�7�

2�6� �73��1� �

53�7�� � �

2�1�2�

�xyz� � �

315

2�1

1

�121�

�1

�6

�17� � �

�1

3

2

�18373

1

�73

�53�� 6

�17�

�3x

x

5x

2y

y

y

z

2z

z

6

�1

7

88.

Solution: ��2, 4, 3�

� ��11�3�1

�6�2�1

�2�1�1� �

12�25

10� � ��2

43�

�xyz� � �

�12

�1

4�9

5

�25

�4��1

�12

�2510�

��x

2x

�x

4y

9y

5y

2z

5z

4z

12

�25

10

89.

Solution: �6, 1, �1�

� ��

59��13� �

19��11� � 1�0�

19��13� �

29��11� � 0�0�

�19��13� �

29��11� � 1�0�

� � �61

�1�

�xyz� � �

�2

�1

0

1

�4

�1

2

1

�1�

�1

��13�11

0� � ��5

919

�19

19

�2929

�1

0

�1���13

�110�

��2x � y

�x � 4y

�y

2z

z

z

�13

�11

0

Review Exercises for Chapter 8 803

90.

Solution: ��3, 5, 0�

�xyz� � �

3�1�8

�114

56

�1��1

��14

844� � �

256496

�23

�196

�37623

116

236

�13

� ��14

844� � �

�350�

�3x

�x

�8x

y

y

4y

5z

6z

z

�14

8

44

91.

Solution: ��3, 1�

�x

y� � �1

3

2

4��1��1

�5� � ��232

1

�12���1

�5� � ��3

1�

� x

3x

2y

4y

�1

�5

92.

Solution: �5, 6)

y � 6x � 5,

�xy� � � 1

�632�

�1

� 23�18� � �0.1

0.3�0.15

0.05� � 23�18� � �5

6�

� x

�6x

3y

2y

23

�18

93.

Solution: �1, 1, �2�

� �11

�2� �xyz� � �

�304

�313

�414�

�1

�2

�1�1� � �

14

�4

04

�3

13

�3��2

�1�1�

��3x

4x

3y

y

3y

4z

z

4z

2

�1

�1

94.

Solution: �4, �2, 1�

z � 1y � �2,x � 4,

�xyz� � �

1�2

1

�37

�1

�23

�3��1

�8

�193� � �

�18�3�5

�7�1�2

511� �

8�19

3� � �4

�21�

�x

�2x

x

3y

7y

y

2z

3z

3z

8

�19

3

95. �82 5

�4� � 8��4��5�2� � �42 96. ��97

11�4� � ��9���4� � �11��7� � �41

97. �50

10

�30

5� � 50�5� � ��30��10� � 550 98. �1412

�24�15� � �14���15� � ��24��12� � 78

804 Chapter 8 Matrices and Determinants

99.

(a)

M22 � 2

M21 � �1

M12 � 7

M11 � 4

�2

7

�1

4�(b)

C22 � M22 � 2

C21 � �M21 � 1

C12 � �M12 � �7

C11 � M11 � 4

100.

(a) (b) C11

C12

C21

C22

M11 � �4

�M12 � �5

�M21 � �6

M22 � 3

M11

M12

M21

M22

�4

5

6

3

�35

6�4�

101.

(a)

(b)

C33 � M33 � 19

C32 � �M32 � 2

C31 � M31 � 5

C23 � �M23 � �22

C22 � M22 � 19

C21 � �M21 � �20

C13 � M13 � �21

C12 � �M12 � 12

C11 � M11 � 30

M33 � � 3

�2

2

5� � 19

M32 � � 3

�2

�1

0� � �2

M31 � �25 �1

0� � 5

M23 � �31 2

8� � 22

M22 � �31 �1

6� � 19

M21 � �28 �1

6� � 20

M13 � ��2

1

5

8� � �21

M12 � ��2

1

0

6� � �12

M11 � �58 0

6� � 30

�3

�2

1

2

5

8

�1

0

6� 102.

(a)

(b)

C33 � M33 � 22

C32 � �M32 � 96

C31 � M31 � �47

C23 � �M23 � �20

C22 � M22 � 32

C21 � �M21 � �2

C13 � M13 � 26

C12 � �M12 � 24

C11 � M11 � 19

M33 � �86 35� � 22

M32 � �86 4�9� � �96

M31 � �35 4�9� � �47

M23 � � 8�4

31� � 20

M22 � � 8�4

42� � 32

M21 � �31 42� � 2

M13 � � 6�4

51� � 26

M12 � � 6�4

�92� � �24

M11 � �51 �92� � 19

�86

�4

351

4�9

2�

103. Expand using Column 2.

� �4��34� � 3�2� � 130

��2

�6

5

4

0

3

1

2

4� � �4��6

5

2

4� � 3��2

�6

1

2�

Review Exercises for Chapter 8 805

104. Expand using Row 3.

� �5�25� � �18� � ��26� � �117

� 42

�5

7�3

1

�14

�1� � �5� 7�3

�14� � 1�42 �1

4� � 1�42 7�3�

105. Expand along Row 1.

� 279

� 3�128 � 5 � 56� � 4��12�

� 3�8�8 � ��8�� � 1�1 � 6� � 2��4 � 24�� � 4�0 � 6�8 � 6� � 0�

�3060 0813

�418

�4

0221� � 3�813 1

8�4

221� � ��4��060 8

13

221�

106. Expand using Row 1, then use Row 3 of each matrix.

� �255

� �5�54 � 3� � 6�9 � 9�

� �5�6��1 � 10� � 3��5 � 4�� � 6���1 � 10� � 3�0 � 3��

��5

0

�3

1

6

1

4

6

0

�1

�5

0

0

2

1

3� � �5�146 �1

�5

0

2

1

3� � 6� 0

�3

1

�1

�5

0

2

1

3�3 � 3

107.

Solution: �4, 7�

y �� 5

�11

6

�23�� 5

�11

�2

3� ��49

�7� 7x �

� 6

�23

�2

3�� 5

�11

�2

3� ��28

�7� 4,

� 5x

�11x

2y

3y

6

�23

108.

Solution: �3, �2�

y ��39 �7

37��39 8

�5� �174�87

� �2x ���7

37

8

�5��39 8

�5� ��261�87

� 3,

�3x

9x

8y

5y

�7

37

806 Chapter 8 Matrices and Determinants

109.

Solution: ��1, 4, 5�

��2(�27) � 4�1� � 1��20�

14�

7014

� 5

z � ��2

4

�1

3

�1

�4

�11

�3

15�14

�2��1�2��1

�4

�3

15� � 4��1�3� 3

�4

�11

15� � 1��1�4� 3

�1

�11

�3�14

��2��33� � 4�9� � 1��26�

14�

5614

� 4

y � ��24

�1

�11�315

�516�

14�

�2��1�2��315

16� � 4��1�3��11

15�5

6� � 1��1�4��11�3

�51�

14

��11��2� � 3��2� � 15��2�

14�

�1414

� �1

x � ��11

�3

15

3

�1

�4

�5

1

6�14

�11��1�2��1�4

16� � 3��1�3� 3

�4�5

6� � 15��1�4� 3�1

�51�

14

� �2��2� � 4��2� � ��2� � 14

D � ��2

4

�1

3

�1

�4

�5

1

6� � �2��1�2��1

�4

1

6� � 4��1�3� 3

�4

�5

6� � 1��1�4� 3

�1

�5

1��

�2x

4x

�x

3y

y

4y

5z

z

6z

�11

�3

15

110.

Solution: �6, 8, 1�

z � �532 �2�3�1

15�7�3�

65�

6565

� 1 y � �532 15�7�3

1�1�7�

65�

52065

� 8, x � � 15�7�3

�2�3�1

1�1�7�

65�

39065

� 6,

D � �532 �2�3�1

1�1�7� � 65�

5x

3x

2x

2y

3y

y

z

z

7z

15

�7

�3

,

111.

Area �1

2�155 0

0

8

1

1

1� �1

21�08 1

1� � 1�55 0

8� �1

2��8 � 40� �

1

2 �32� � 16 square units

�1, 0�, �5, 0�, �5, 8�

Review Exercises for Chapter 8 807

114.

Area square units�1

2� 32

4

4

1� 1

2

2

1

1

1� �1

225

4 �25

8

�32, 1�, �4, � 1

2�, �4, 2� 115.

The points are collinear.

��13

�3

7�915

111� � 0

��1, 7�, �3, �9�, ��3, 15�

117.

x � 2y � 4 � 0

�4x � 8y � 16 � 0

�16 � �4x � 4y� � 4y � 0

1��44

04� � 1�x

4y4� � 1� x

�4y0� � 0

� x

�4

4

y

0

4

1

1

1� � 0

��4, 0�, �4, 4�

118.

3x � 2y � 16 � 0

6x � 4y � 32 � 0

�x

2

6

y

5

�1

1

1

1� � 0

�2, 5�, �6, �1� 119.

2x � 6y � 13 � 0

�13 � �x �72y� � �3x �

52y� � 0

1��5272

3

1� � 1�x72 y1� � 1� x

�52

y

3� � 0

� x

�5272

y

3

1

1

1

1� � 0

��52, 3�, �7

2, 1�

120.

Multiply both sides by

10x � 5y � 9 � 0

�103 .�3x � 1.5y � 2.7 � 0

� x

�0.8

0.7

y

0.2

3.2

1

1

1� � 0

��0.8, 0.2�, �0.7, 3.2�

116. Points:

The points are collinear.

� 50 � 40 � 10 � 0

� 0�2

8

�5�6�1

111� � ��2

8�6�1� � �08 �5

�1� � � 0�2

�5�6�

�0, �5�, ��2, �6�, �8, �1�

112.

Area square units�1

2��4

4

0

0

0

6

1

1

1� �1

2�48� � 24

��4, 0�, �4, 0�, �0, 6� 113.

� �1

2��5�3� � ��5�� � 10 square units

� �1

2�5� 1

�2

1

1� � � 1

�2

�4

3�Area � �

1

2� 1

�2

0

�4

3

5

1

1

1��1, �4�, ��2, 3�, �0, 5�

808 Chapter 8 Matrices and Determinants

121.

Cryptogram:�42 �60 �53 20 21 99 �30 �69�21 6 0 �68 8 45 102

�15 23 0��23

�6

�202

0�3

3� � �99 �30 �69�

�2 5 12��23

�6

�202

0�3

3� � ��53 20 21�

�21 20 0��23

�6

�202

0�3

3� � �102 �42 �60�

�11 0 15��2

3

�6

�2

0

2

0

�3

3� � ��68 8 45�

�12 15 15��23

�6

�202

0�3

3� � ��21 6 0�

A � �2

3

�6

�2

0

2

0

�3

3�

L[12

O15

O15]

K[11

__0

O15]

U[21

T20

__0]

B[2

E5

L12]

O[15

W23

__0]

122. R E T U R N __ T 0 __ B A S E __

Cryptogram: 66 28 10 �24 �59 �22 �75 �90 �25 �9 �10 �3 8 �11 �10

�19 5 0� A � �8 �11 �10�

�0 2 1� A � ��9 �10 �3�

�0 20 15� A � ��75 �90 �25�

�21 18 14� A � ��24 �59 �22�

�18 5 20� A � �66 28 10�

A � �2

�63

1�6

2

0�2

1��18 5 20� �21 18 14� �0 20 15� �0 2 1� �19 5 0�

Review Exercises for Chapter 8 809

123.

Message: SEE YOU FRIDAY

�245 �171 147���1

24

21

�2

�305� � �1 25 0� A Y __

�32 �15 20���1

2

4

2

1

�2

�3

0

5� � �18 9 4� R I D

��57 48 �33���1

24

21

�2

�305� � �21 0 6� U __ F

�370 �265 225���1

24

21

�2

�305� � �0 25 15� __ Y O

��5 11 �2���1

2

4

2

1

�2

�3

0

5� � �19 5 5� S E E

A�1 � ��1

2

4

2

1

�2

�3

0

5�

124.

Message: MAY THE FORCE BE WITH YOU

M__EOEEI

__U

AT__R____TY__

YHFCBWHO__

1305

155590

21

1200

1800

20250

258632

238

150

14526423

129�9159219370

�105

�105�188�16�84

8�118�152�265

84

921601578

�5100133225

�63

��1

24

21

�2

�305� �

A�1 � ��1

24

21

�2

�305�

125. False. The matrix must be square.

126. True. Expand along Row 3.

Note: Expand each of these matrices along Row 3 to see the previous step.

� �a11

a21

a31

a12

a22

a32

a13

a23

a33� � �a11

a21

c1

a12

a22

c2

a13

a23

c3� � c1 �a12

a22

a13

a23� � c2 �a11

a21

a13

a23� � c3 �a11

a21

a12

a22� � a31�a12

a22

a13

a23� � a32 �a11

a21

a13

a23� � a33 �a11

a21

a12

a22� � �a31 � c1� �a12

a22

a13

a23� � �a32 � c2� �a11

a21

a13

a23� � �a33 � c3� �a11

a21

a12

a22�� a11

a21

a31 � c1

a12

a22

a32 � c2

a13

a23

a33 � c3�

810 Chapter 8 Matrices and Determinants

131.

� � �3 ± 2�10

� ��6 ± �36 � 4��31�

2

�2 � 6� � 31 � 0

�16 � 6� � �2 � 15 � 0

�2 � ����8 � �� � 15 � 0

�2 � �

3

5

�8 � �� � 0

Problem Solving for Chapter 8

1.

(a)

Original Triangle AT Triangle AAT Triangle

The transformation A interchanges the x and y coordinates and then takes the negative of the x coordinate. A represents a counterclockwise rotation by

(b)

represents a clockwise rotation by 90�.A�1

A�1 � � 0�1

10�

A�1�AT� � �A�1A�T � IT � T

A�1�AAT� � �A�1A��AT� � �I��AT� � AT

90�.

y

x1−1

−4

−3

−2

1

2

3

4

−2−3−4 2 3 4

(−2, −4)

(−3, −2)(−1, −1)

y

x1−1

−4

−3

−2

1

2

3

4

−2−3−4 2 3 4

(−2, 3)(−4, 2)

(−1, 1)

y

x1−1

−4

−3

−2

1

2

3

4

−2−3−4 2 3 4

(2, 4)

(3, 2)

(1, 1)

AT � ��11

�42

�23� AAT � ��1

�1�2�4

�3�2�

A � �01

�10� T � �1

124

32�

127. The matrix must be square and its determinant nonzero to have an inverse.

129. No. Each matrix is in row-echelon form, but the thirdmatrix cannot be achieved from the first or secondmatrix with elementary row operations. Also, the firsttwo matrices describe a system of equations with onesolution. The third matrix describes a system withinfinitely many solutions.

128. If A is a square matrix, the cofactor of the entry iswhere is the determinant obtained by

deleting the ith row and jth column of A. The determinantof A is the sum of the entries of any row or column of Amultiplied by their respective cofactors.

Mij��1�i� jMi j,aijCij

130. The part of the matrix corresponding to the coefficientsof the system reduces to a matrix in which the numberof rows with nonzero entries is the same as the numberof variables.

Problem Solving for Chapter 8 811

3. (a)

A is idempotent.

(b)

A is not idempotent.

A2 � �01

10��

01

10� � �1

001� � A

A2 � �10

00��

10

00� � �1

000� � A (c)

A is not idempotent.

(d)

A is not idempotent.

A2 � �21

32��

21

32� � �7

4127� � A

A2 � � 2�1

3�2��

2�1

3�2� � �1

001� � A

4.

(a)

(b)

Thus,

(c)

Thus, A�1 �15

�2 I � A�.

15

�2I � A�A � I

�15

�A � 2I�A � I

�A � 2I�A � �5I

A2 � 2A � �5I

A2 � 2A � 5I � 0

A�1 �15

�2 I � A�.

15

�2 I � A� �15��

20

02� � � 1

�221�� �

15�

12

�21�

A�1 �1

�1� � ��4��12

�21� �

15�

12

�21�

� �00

00� � 0

� ��3�4

4�3� � ��2

4�4�2� � �5

005�

A2 � 2A � 5I � � 1�2

21��

1�2

21��2� 1

�221� � 5�1

001�

A � � 1�2

21�

2. (a) 20000–17 18–64 65+

20150–17 18–64 65+

�4.06%5.12%8.36%1.69%4.81%

10.99%13.23%22.25%4.07%

10.74%

2.63%3.26%5.63%1.05%2.12%

� Northeast Midwest South Mountain Pacific

�4.64%5.91%9.09%1.75%4.30%

11.79%14.03%22.11%3.98%9.96%

2.62%2.94%4.42%0.72%1.74%

� Northeast Midwest South Mountain Pacific

(b) Change in Percent of Populationfrom 2000 to 2015

0–17 18–64 65+

(c) All regions show growth in the 65+ age bracket,especially the South. The South, Mountain and Pacificregions show growth in the 18–64 age bracket. Only thePacific region shows growth in the 0–17 age bracket.

��0.58%�0.79%�0.73%�0.06%

0.51%

�0.80%�0.80%

0.14%0.09%0.78%

0.01%0.32%1.21%0.33%0.38%

� Northeast Midwest South Mountain Pacific

812 Chapter 8 Matrices and Determinants

8. From Exercise 3 we have the singular matrix

where

Also, has this property.A � �10

10�

A2 � A.A � �10

00�

6.

If then

Equating the first entry in Row 1 yields

Now check in the other entries:

Thus, x � 4.

3�9 � 2�4� � �3

2�9 � 2�4� � �2

�4�9 � 2�4� � 4

x � 4

�3�9 � 2x

� 3 ⇒ �3 � �27 � 6x ⇒ x � 4.

� � 3�2

x�3�.�

�3�9 � 2x�A � A�1,

A � � 3�2

x�3� ⇒ A�1 �

1�9 � 2x

��32

�x3�

2�9 � 2x

�x�9 � 2x

3�9 � 2x

7. If is singular then

Thus, x � 6.

ad � bc � �12 � 2x � 0.

A � � 4�2

x�3�

9.

Thus, �1aa2

1bb2

1cc2� � �a � b��b � c��c � a�.

�1aa2

1bb2

1cc2� � �bb2

cc2� � �aa2

cc2� � �aa2

bb2� � bc2 � b2c � ac2 � a2c � ab2 � a2b

�a � b��b � c��c � a� � �a2b � a2c � ab2 � ac2 � b2c � bc2

5. (a)

Gold Cable Company: 28,750 households

Galaxy Cable Company: 35,750 households

Nonsubscribers: 35,500 households

� �28,75035,75035,500��

25,00030,00045,000��

0.700.200.10

0.150.800.05

0.150.150.70� (b)

Gold Cable Company: 30,813 households

Galaxy Cable Company: 39,675 households

Nonsubscribers: 29,513 households

� �30,81339,67529,513��

28,75035,75035,500��

0.700.200.10

0.150.800.05

0.150.150.70�

(c)

Gold Cable Company: 31,947 households

Galaxy Cable Company: 42,329 households

Nonsubscribers: 25,724 households

� �31,94742,32925,724��

30,812.539,675

29,512.5��0.700.200.10

0.150.800.05

0.150.150.70�

(d) Both cable companies are increasing the number of subscribers, while the number of nonsubscribers isdecreasing each year.

Problem Solving for Chapter 8 813

12.

From Exercise 11

� ax3 � bx2 � cx � d

� x�1

00

0x

�10

00x

�1

dcba � � x� x

�10

0x

�1

cba � � d��1

00

x�1

0

0x

�1� � x�ax2 � bx � c� � d� � ��10

x�1�

13.

Element Atomic mass

Sulfur 32

Nitrogen 14

Fluoride 19

S � �184146104

402

064�

�64�

�2048�64

� 32

D � �410 402

064� � �64

2N � 4F � 104

S � 6F � 146

4S � 4N � 184

F � �410 402

184146104�

�64�

�1216�64

� 19

N � �410 184146104

064�

�64�

�896�64

� 14

14. Let cost of a transformer, cost per foot of wire, cost of a light.

By using the matrix capabilities of a graphing calculator to reduce the augmented matrix to reduced row-echelon form, we have the following costs:

Transformer $10.00

Foot of wire $ 0.20

Light $ 1.00

�100

010

001

���

100.2

1�→rref�

111

2550

100

51520

���

203550�

x � 100y � 20z � 50

x � 50y � 15z � 35

x � 25y � 5z � 20

z �y �x �

10.

Thus, �1aa3

1bb3

1cc3� � �a � b��b � c��c � a��a � b � c�.

�1aa3

1bb3

1cc3� � �bb3

cc3� � �aa3

cc3� � �aa3

bb3� � bc3 � b3c � ac3 � a3c � ab3 � a3b

�a � b��b � c��c � a��a � b � c� � �a3b � a3c � ab3 � ac3 � b3c � bc3

11. � ax2 � bx � c� x�ax � b� � c�1 � 0� � x�1

0

0x

�1

cba� � x� x

�1ba� � c��1

0x

�1�

814 Chapter 8 Matrices and Determinants

16.

0 18 5 13 5 13 2 5 18 0

__ R E M E M B E R __

19 5 16 20 5 13 2 5 18 0

S E P T E M B E R __

20 8 5 0 5 12 5 22 5 14 20 8 0

T H E __ E L E V E N T H __

REMEMBER SEPTEMBER THE ELEVENTH

2331252441203813224128

13�34�17

14�17�29�56�11�3

�53�32

�346361

�37�840

1161

�68516

�111

�711

�211

611211

�1

11

411511311

� �

01320

16131885

2220

1855

1920205

1258

51318555

2005

140

A � �111

�21

�1

2�3

4� ⇒ A�1 � �111

�711

�211

611211

�1

11

4115

113

11

15.

Thus, �AB�T � BTAT.

BTAT � ��30

12

1�1��

�11

�2

201� � �2

4�5�1�

�AB�T � �24

�5�1�AB � � 2

�54

�1�,

BT � ��30

12

1�1�AT � �

�11

�2

201�,

B � ��3

11

02

�1�A � ��12

10

�21�,

Problem Solving for Chapter 8 815

18.

Conjecture: �A�1� �1

�A�

�A� � 16 and �A�1� �1

16

A�1��1

16316

�18

�7161116

�18

58

�9834�

A � �601

421

132� 19. Let then

Let then

Let then

Conjecture: If A is an matrix, each of whose rowsadd up to zero, then .�A� � 0

n � n

�A� � 0.A � �3

�659

�748

11

50

�6�4

�12

�7�16

�,

�A� � 0.A � �2

�35

41

�8

�623�,

�A� � 0.A � �35

�3�5�,

20. (a) Answers will vary.

(b) for n an integer

for n an integer

(c) if A is

(d) Conjecture: If A is then An � 0.n � n,

4 � 4.A4 � 0

≥ 3.B3 � 0 so Bn � 0

B2 � �000

000

2800�

≥ 2.A2 � 0 so An � 0

B � �000

400

�170�A � �0

030�,

17. (a)

A�1 � �11

�2�3�

45x � 35z � 1538x � 30z � 14� ⇒ x � �2, z � �3

45w � 35y � 1038w � 30y � 8� ⇒ w � 1, y � 1

38x � 30z � 14

38w � 30y � 8

45x � 35z � 15

45w � 35y � 10

�38 �30 �wy

xz� � �8 14

�45 �35 �wy

xz� � �10 15

(b)

JOHN RETURN TO BASE

453818358142752

2215

�35�30�18�30�60�28�55�2

�21�10

�11

�2�3� �

10805

211420015

15141820180

152

190

JH

—EUNT

—AE

ONRTR

—OBS

Chapter 8 Practice Test

816 Chapter 8 Matrices and Determinants

1. Put the matrix in reduced row-echelon form.

�1

3

�2

�5

4

9�

For Exercises 2–4, use matrices to solve the system of equations.

2.

2x � 5y � �11�3x � 5y � �13

5. Multiply �1

2

4

0

5

�3� �1

0

�1

6

�7

2�.

6. Given find 3A � 5B.A � � 9

�4

1

8� and B � �6

3

�2

5�,

7. Find

f �x� � x2 � 7x � 8, A � �3

7

0

1�f �A�.

8. True or false:

where and are matrices.

(Assume that exist.)A2, AB, and B2

BA�A � B��A � 3B� � A2 � 4AB � 3B2

9. �1

3

2

5� 10. �1

3

6

1

6

10

1

5

8�

11. Use an inverse matrix to solve the systems.

(a) (b)

3x � 5y � �23x � 5y � 1

3x � 2y � �33x � 2y � 4

For Exercises 12–14, find the determinant of the matrix.

12. �6

3

�1

4� 13. �1

5

6

3

9

2

�1

0

�5� 14. �

1

0

3

2

4

1

5

0

2

�2

�1

6

3

0

1

1�

3.

3x � 2y � �1�3x � 2y � �8

2x � 3y � �3 4.

3x � y � 3z � �3�2x � y � 3z � �0

2x � y � 3z � �5

For Exercises 9–10, find the inverse of the matrix, if it exists.

Practice Test for Chapter 8 817

16. Use a determinant to find the area of the triangle withvertices and �3, 9�.�0, 7�, �5, 0�,

17. Find the equation of the line through �2, 7� and ��1, 4�.

For Exercises 18–20, use Cramer’s Rule to find the indicated value.

18. Find x.

�6x

2x

7y

5y

4

11

19. Find z.

�3x

x

y

y

z

4z

1

3

2

20. Find y.

�721.4x

45.9x

29.1y

105.6y

33.77

19.85

15. Evaluate ��60000 4

5

0

0

0

3

1

2

0

0

0

4

7

9

0

6

8

3

2

1� .