CHAPTER 8 Matrices and Determinants - Saddleback College · CHAPTER 8 Matrices and Determinants ......
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�
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49.
Cryptogram: �5 �41 �87 91 207 257 11 �5 �41 40 80 84 76 177 227
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Cryptogram: 58 122 139 1 �37 �95 40 67 55 23 17 �19 47 88 88 14 21 11
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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
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54.
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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
__
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I
H
__
R
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1
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16��1
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15
18
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15
18
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1
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15
18
14�
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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� .