Solve the problem.
64)
Let A =
–1 2 –7
8 7 4
–6 2 2
and B =
1–8 2
–2–4–5
–3 9 6
. Find –3A + 2B.
64)
A)
0 6 –9
–6 3 11
–5–1 8
B)
0–6–5
6 3 –1
–911 8
C)
4–14 23
–26 –25 –17
15 3 0
D)
5–22 25
–28 –29 –22
12 12 6
65)
Let A =
2–6 3
4–8–2
–5 5 6
and B =
7 6 2
–5 0 –3
4 7 –5
. Find A – B.
65)
A)
–512 1
9–8 1
–9 2 11
B)
–5–12 1
9–8 1
–9–211
C)
–5 0 1
9–8 1
1–2 1
D)
9 0 5
–1–8–5
–112 1
Use Cramer’s rule to determine if the system is inconsistent system or contains dependent equations.
66)
x + z = 1
2x – 2y = – 2
y + z = 4
66)
A)
system contains dependent equations
B)
system is inconsistent
Solve the system of equations using matrices. Use Gauss–Jordan elimination.
67)
x = 4 – y – z
x – y + 2z= – 6
2x + y = 9 – z
67)
A)
{(–4, 3, 5)}
B)
{(3, –4, 5)}
C)
{(5, 3, –4)}
D)
{(–4, 5, 3)}
Solve the problem.
68)
Let A =–3 3
0 2 . Find 2A.
68)
A)
–6 3
0 2
B)
–1 5
2 4
C)
–6 6
0 2
D)
–6 6
0 4
Solve the system of equations using matrices. Use Gaussian elimination with back–substitution.
69)
x – y + 3z= 2
3x + z =2
x + 4y + z =18
69)
A)
{(2, 4, 0)}
B)
{(0, 4, 2)}
C)
{(0, 2, 4)}
D)
{(2, 0, 4)}
Solve the system of equations using matrices. Use Gauss–Jordan elimination.
70)
8x –y– 4z=20
5x + 7y+ 3z=101
9x – 6y+z=11
70)
A)
{(12, 8, –6)}
B)
{(6, 5, 8)}
C)
{(–6, 8, 12)}
D)
{(6, 8, 5)}
Evaluate the determinant.
71)
5 3 4 5
1 9 6 2
4 7 0 5
6 9 3 9
71)
A)
20
B)
–22
C)
21
D)
–501
Use Cramer’s rule to solve the system.
72)
–2x + 3y= – 5
4x + 3y=19
72)
A)
{(4, –1)}
B)
{(4, 1)}
C)
{(1, 4)}
D)
{(–1, 4)}
Write a system of linear equations in three variables, and then use matrices to solve the system.
73)
A ceramics workshop makes wreaths, trees, and sleighs for sale at Christmas. A wreath takes 3
hours to prepare, 2 hours to paint, and 8 hours to fire. A tree takes 15 hours to prepare, 3 hours to
paint, and 4 hours to fire. A sleigh takes 4 hours to prepare, 15 hours to paint, and 7 hours to fire. If
the workshop has 118 hours for prep time, 93 hours for painting, and 120 hours for firing, how
many of each can be made?
73)
A)
4 wreaths; 9 trees; 5 sleighs
B)
9 wreaths; 5 trees; 4 sleighs
C)
5 wreaths; 4 trees; 9 sleighs
D)
10 wreaths; 6 trees; 5 sleighs
Find the product AB, if possible.
74)
A =1 3 –1
3 0 5 , B =
3 0
–1 1
0 5
74)
A)
AB is not defined.
B)
3–3 0
0 0 25
C)
–2 0
25 9
D)
0–2
925
Solve the problem using matrices.
75)
The figure below shows the intersection of three one–way streets. To keep traffic moving, the
number of cars per minute entering an intersection must equal the number of cars leaving that
intersection. Set up a system of equations that keeps traffic moving, and use Gaussian elimination
to solve the system. If construction limits z to t cars per minute, how many cars per minute must
pass through the other intersections to keep traffic moving?
75)
A)
t + 1 cars/min between I2 and I1; t + 4 cars/min between I1 and I3
B)
t – 2 cars/min between I2 and I1; t + 1 cars/min between I1 and I3
C)
t + 8 cars/min between I2 and I1; t + 3 cars/min between I1 and I3
D)
t + 2 cars/min between I2 and I1; t – 3 cars/min between I1 and I3
76)
The final grade for an algebra course is determined by grades on the midterm and final exam. The
grades for four students and two possible grading systems are modeled by the following matrices.
Midterm Final
Student 1
Student 2
Student 3
Student 4
73 79
44 62
85 90
98 96
System
1
System
2
Midterm
Final
0.3 0.5
0.7 0.5
Find the final course score for Student 3 for both grading System 1 and System 2.
76)
A)
System 1: 70.5; System 2: 104.5
B)
System 1: 44.2; System 2: 53
C)
System 1: 88.5; System 2: 87.5
D)
System 1: 77.2; System 2: 76
Solve the problem.
77)
Let A =
2–8
–8 1
6–7
and B =
–2–8
–3–1
–4–8
. Find A + B.
77)
A)
4 0
–5 2
10 2
B)
0–16
11 1
215
C)
0–16
–11 0
2–15
D)
0 1
–11 0
2–15
Evaluate the determinant.
78)
125
254
125
78)
A)
1
B)
0
C)
–16
D)
106
Find the inverse of the matrix, if possible.
79)
A =3–2
2 6
79)
A)
–1
11
3
22
3
11
1
11
B)
3
11
1
11
–1
11
3
22
C)
3
11 –1
11
1
11
3
22
D)
3
22
1
11
–1
11
3
11
Encode or decode the given message, as requested, numbering the letters of the alphabet 1 through 26 in their usual order.
80)
Use the coding matrix A =–1–3
2 5 to encode the message CARE.
80)
A)
18 105
–7–4
B)
–1–8
–4–29
C)
–6–33
11 61
D)
–57 –16
96 27
Evaluate the determinant.
81)
554
114
352
81)
A)
212
B)
–32
C)
–152
D)
32
Use Cramer’s rule to solve the system.
82)
9x + 5y – z =90
x + 9y + 9z =168
7x + y + z =60
82)
A)
{(6, –9, –9)}
B)
{(7, 7, 9)}
C)
{(6, 9, 9)}
D)
{(9, 9, 9)}
26
Find the product AB, if possible.
83)
A =–2 3
3 2 , B =–2 0
–1 1
83)
A)
4–6
–4–1
B)
1 3
–8 2
C)
3 1
2–8
D)
4 0
–3 2
Evaluate the determinant.
84)
–4 1 2
–1 0 –2
5 0 1
84)
A)
–9
B)
11
C)
9
D)
–11
Solve the problem using matrices.
85)
The nutritional content per ounce for three foods is given in the table below.
Fat (g/oz) Protein (g/oz) Fiber (g/oz)
Food A 2 4 1
Food B 1 2 1
Food C 8 16 5
What combination of these foods can provide exactly 14 grams of fat, 27 grams of protein, and 10
grams of fiber?
85)
A)
4 oz of Food A; 6 oz of Food B; 2 oz of Food C
B)
No possible combination of these foods
C)
3 oz of Food A; 5 oz of Food B; 1 oz of Food C
D)
7 oz of Food A; 7 oz of Food B; 1 oz of Food C
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
86)
x + y + z + w =7
3x – 2z + 5w =11
–4x + 3y + w =4
–x – y – z – w =6
86)
A)
{(–11, 7
19 , 6
19 , –4)}
B)
{(7
4, –1
2, 5, –1
6)}
C)
{(3
2, 1, 1
3, –2)}
D)
Find the inverse of the matrix, if possible.
87)
A =
1 2 0 0
0 1 2 0
0 0 1 –9
0 0 0 1
87)
A)
1 0 0 0
–2 1 0 0
4–2 1 0
–18 –18 9 1
B)
1 9 18 36
0 1 –2–4
0 0 1 –2
0 0 0 1
C)
1–2 4 36
0 1 –2–18
0 0 1 9
0 0 0 1
D)
1 0 0 0
9 1 0 0
18 –2 1 0
36 –4–2 1
Find the product AB, if possible.
88)
A =–1 3
1 4 , B =0–2 5
1–3 2
88)
A)
0–615
1–12 8
B)
AB is not defined.
C)
3–7 1
4–14 13
D)
3 4
–7–14
113
Use Cramer’s rule to solve the system.
89)
4x – 6z =4
–3x + 2y – 5z = – 20
6x – 2y =22
89)
A)
{(1, 2, 1)}
B)
{(4, –1, –2)}
C)
{(5, –1, 2)}
D)
{(4, 1, 2)}
Give the order of the matrix, and identify the given element of the matrix.
90)
6 8 7 –7
–9–1 4 0 ; a12
90)
A)
4 × 2; –9
B)
4 × 2; 8
C)
2 × 4;–9
D)
2 × 4; 8
Use Cramer’s rule to determine if the system is inconsistent system or contains dependent equations.
91)
4x – y + 2z = 1
3x + 5y – z = 0
–6x – 10y + 2z = 0
91)
A)
system is inconsistent
B)
system contains dependent equations
29
Use Cramer’s rule to solve the system.
92)
5x + 4y – z =28
x – 3y + 4z =32
2x + y + z =22
92)
A)
{(3, 9, 3)}
B)
{(5, 3, 9)}
C)
{(6, 1, 9)}
D)
{(5, –3, –9)}
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
93)
x + 3y + 2z =11
4y + 9z = – 12
x + 7y + 11z = – 1
93)
A)
{(19z
4+ 20, 9z
4+ 3, z)}
B)
{(19z
4+ 20, –9z
4+ 3, z)}
C)
{(19z
4+ 20, –9z
4– 3, z)}
D)
{(–19z
4+ 20, –9z
4+ 3, z)}
Use Cramer’s rule to determine if the system is inconsistent system or contains dependent equations.
94)
4x +y=14
12x+ 3y=42
94)
A)
system contains dependent equations
B)
system is inconsistent
Write the linear system as a matrix equation in the form AX = B, where A is the coefficient matrix and B is the constant
matrix.
95)
3x + 5y + 7z = 53
9x + 8y + 3z = 19
–2x – 2y – 2z = – 14
95)
A)
3 9 –2
5 8 –2
7 3 –2
x
y
z
=
53
19
–14
B)
3 5 7
9 8 3
–2–2–2
x
y
z
=
53
19
–14
C)
53 7 5
19 3 8
–14 –2–2
x
y
z
=
3
9
–2
D)
3 5
9 8
–2–2
x
y
z
=
7
3
–2
30
The shape in the figure below is shown using 9 pixels in a 3 ×
3 grid. The color levels are given to the right of the
figure. Use the matrix
1 3 1
1 3 1
3 3 3
that represents a digital photograph of the shape to solve the problem.
96)
Adjust the contrast by changing the black to light grey and the light grey to black. Use matrix
addition to accomplish this.
96)
A)
1 3 1
1 3 1
3 3 3
+
1–1 1
1–1 1
–1–1–1
=
3 1 3
3 1 3
1 1 1
B)
131
131
333
+
2 2 2
2 2 2
2 2 2
=
3 1 3
3 1 3
1 1 1
C)
1 3 1
1 3 1
3 3 3
+
–2–2–2
–2–2–2
–2–2–2
=
3 1 3
3 1 3
1 1 1
D)
131
131
333
+
2 –2 2
2 –2 2
–2–2–2
=
3 1 3
3 1 3
1 1 1
Solve the problem using matrices.
97)
A company that manufactures products A, B, and C does both assembly and testing. The hours
needed to assemble and test each product are shown in the table below.
Hours needed
weekly to assemble
Hours needed
weekly to test
Product A 1 4
Product B 1 5
Product C 2 10
The company has exactly 21 hours per week available for assembly and 95 hours per week
available for testing. If the company must produce t units of Product C this week, how many units
of Products A and B can they produce?
97)
A)
10 of Product A; –2t +11 of Product B
B)
10 of Product A; 11 of Product B
C)
10t of Product A; 2t +11 of Product B
D)
t +10 of Product A; t +11 of Product B
Write a system of linear equations in three variables, and then use matrices to solve the system.
98)
The table below shows the number of birds for three selected years after an endangered species
protection program was started.
x (Number of years after 1980) 1 5 10
y (Number of birds) 42 202 627
Use the quadratic function y = ax2+ bx + c to model the data. Solve the system of linear equations
involving a, b, and c using matrices. Find the equation that models the data.
98)
A)
y =10x2– 30x +23
B)
y =6x2+20x +22
C)
y =7x2– 10x +30
D)
y =5x2+10x +27
Use Cramer’s rule to solve the system.
99)
3x = – 7+ 2y
4y =26 + 2x
99)
A)
{(3, 8)}
B)
{(–8, 3)}
C)
{(3, –8)}
D)
{(8, 3)}
32
The shape in the figure below is shown using 9 pixels in a 3 ×
3 grid. The color levels are given to the right of the
figure. Use the matrix
1 3 1
1 3 1
3 3 3
that represents a digital photograph of the shape to solve the problem.
100)
Adjust the contrast by changing the black to dark grey and the light grey to white. Use matrix
addition to accomplish this.
100)
A)
1 3 1
1 3 1
3 3 3
+
–1–1–1
–1–1–1
–1–1–1
=
0 2 0
0 2 0
2 2 2
B)
131
131
333
+
0 –1 0
0 –1 0
–1–1–1
=
0 2 0
0 2 0
2 2 2
C)
1 3 1
1 3 1
3 3 3
+
1 1 1
1 1 1
1 1 1
=
2 4 2
2 4 2
4 4 4
D)
131
131
333
+
0 –1 0
0 –1 0
–1–1–1
=
1 2 1
1 2 1
2 2 2
101)
Using the same color levels from the instructions, write a 3 × 3 matrix A that represents the letter L
in dark grey on a white background. Then find a 3 × 3 matrix B so that A + B lightens only the letter
L from dark grey to light grey.
101)
A)
A =
1 0 0
1 0 0
1 1 1
; B =
–1 0 0
–1 0 0
–1–1–1
B)
A =
3 1 1
3 1 1
3 3 3
; B =
–2–1–1
–2–1–1
–2–2–2
C)
A =
2 0 0
2 0 0
2 2 2
; B =
–1 0 0
–1 0 0
–1–1–1
D)
A =
2 1 1
2 1 1
2 2 2
; B =
100
100
111
Find the product AB, if possible.
102)
A = [6–1–1], B =
9–4 6
6–9–4
–1 5 –9
102)
A)
6–1 0 –1
9–4 6
6–9–4
–1 5 –9
B)
49 –20 49
C)
49
–20
49
D)
54 4–6
36 9 4
–6–5 9
Evaluate the determinant.
103)
0 0 0 7
4 4 4 9
6 1 3 8
2 5 5 9
103)
A)
–35
B)
168
C)
–168
D)
–21
Find values for the variables so that the matrices are equal.
104)
1 7
9 –8
=x y
9 z
104)
A)
x =1; y =7; z =9
B)
x =1; y =9; z = – 8
C)
x =1; y =7; z = – 8
D)
x =7; y =1; z = – 8
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
105)
x + y + z + w = 8
3x + 2y + z + 4w = 21
4x + 4y + 5z + 8w = 30
2x + 3y + 6z + 9w = 15
105)
A)
B)
{(5w + 11, –3w – 7, –3w + 4, w)}
C)
{(–3, 16, –6, 1)}
D)
{(–6w + 3, 9w + 7, –4w – 2, w)}
Use Cramer’s rule to solve the system.
106)
2x + 3y=34
2x –2y =4
106)
A)
{(6, 8)}
B)
{(–8, –6)}
C)
{(–6, 8)}
D)
{(8, 6)}
Solve the problem.
107)
Let A = [–2 2] and B = [1 0]. Find 2A + 3B.
107)
A)
1 2
B)
–3 4
C)
–1 4
D)
–4 4
35
Write the augmented matrix for the system of equations.
108)
8x +9y –4z + w =3
10y + z =12
x – y –12z =8
2x –2y +6z = – 4
108)
A)
8 9 –4 3
010 1 6
1–1–12 8
2–2 6 –4
B)
8 9 4 1 3
010 1 0 12
1 1 12 0 8
2 2 6 0 –4
C)
8 9 –4 1 3
010 1 0 12
1–1–12 0 8
2–2 6 0 –4
D)
8 0 1 9
910 –1–2
–4 1 –12 6
1 0 0 0
312 8–4
Encode or decode the given message, as requested, numbering the letters of the alphabet 1 through 26 in their usual order.
109)
Use the coding matrix A =2 1
5 3 and its inverse A–1= 3 –1
–5 2 to decode the cryptogram 9 6
25 17 .
109)
A)
BEAD
B)
CURB
C)
CARE
D)
DARE
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
110)
5x + 2y + z = – 11
2x – 3y – z =17
7x – y =12
110)
A)
{(0, –6, 1)}
B)
C)
{(–2, 0, –1)}
D)
{(1, –5, 0)}
36
Write the matrix equation as a system of linear equations without matrices.
111)
6 3
5 4
x
y
=–3
4
111)
A)
6x + 3y = – 3
4x + 5y = 4
B)
6x + 3y = – 3
5x + 4y = 4
C)
3x + 6y = – 3
5x + 4y = 4
D)
6x + 3y =3
5x + 4y = – 4
Solve the problem.
112)
Let A =
2–10 3
4–12 –1
–4 4 3
and B =
610 1
–4 0 –5
3 5 –4
. Find A + B.
112)
A)
8 0 4
0–12 –6
–1 9 –1
B)
020 4
8–12 –6
–1 9 –1
C)
8 0 4
8–12 4
1 9 –1
D)
020 4
0–12 4
1 9 –1
Find the product AB, if possible.
113)
A =6–2–8
–4 1 8 , B =
–2
–2
6
113)
A)
–56
54
B)
6–2–8
–4 1 8
–2–2 6
C)
–56 54
D)
AB is not defined.