Evaluate the determinant.
59)
125
254
125
59)
A)
–16
B)
106
C)
0
D)
1
60)
–8–4
–4 6
60)
A)
–32
B)
56
C)
64
D)
–64
Find values for the variables so that the matrices are equal.
61)
x y +8
2z 1
=10 13
18 1
61)
A)
x =13; y =1; z =10
B)
x =10; y =5; z =9
C)
x =10; y =13; z =18
D)
x =1; y =21; z =36
62)
1 7
9 –8
=x y
9 z
62)
A)
x =1; y =7; z = – 8
B)
x =1; y =7; z =9
C)
x =7; y =1; z = – 8
D)
x =1; y =9; z = – 8
21
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
63)
3x + y + z – 2w = 10
2x + 3y + 3z + w = – 5
2x + y + 4z + 11w = 11
63)
A)
{(7, –1, –6, 2)}
B)
{(w + 5, 3w – 7, –4w + 2, w)}
C)
{(6, –4, –2, 1)}
D)
{(2w + 3, 6w – 7, –10w + 8, w)}
Encode or decode the given message, as requested, numbering the letters of the alphabet 1 through 26 in their usual order.
64)
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 .
64)
A)
DARE
B)
CURB
C)
BEAD
D)
CARE
Perform the matrix row operation (or operations) and write the new matrix.
65)
3–4 1 2
–5 0 2 –2
–1 4 –3–1
–3R1+R2
65)
A)
–14 12 –1–8
–5 0 2 –2
–1 4 –3–1
B)
3–4 1 2
–14 12 –1–8
–1 4 –3–1
C)
18 –4–5 8
–5 0 2 –2
–1 4 –3–1
D)
3–4 1 2
4–12 5 4
–1 4 –3–1
22
Evaluate the determinant.
66)
0 7 9 8
0 4 9 4
4 7 2 4
5 1 6 6
66)
A)
1280
B)
20
C)
–36
D)
6
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
67)
x + y + z = 7
x – y + 2z =7
2x + 3z =14
67)
A)
{(–3z
2– 7, 2z, z)}
B)
{(–3z
2+ 7, z
2, z)}
C)
{(–3z
2+ 7, 2z, z)}
D)
{(–3z
2– 7, z
2, z)}
Evaluate the determinant.
68)
554
114
352
68)
A)
–152
B)
32
C)
–32
D)
212
23
Solve the problem.
69)
Let A =
2–10 3
4–12 –1
–4 4 3
and B =
610 1
–4 0 –5
3 5 –4
. Find A + B.
69)
A)
8 0 4
0–12 –6
–1 9 –1
B)
8 0 4
8–12 4
1 9 –1
C)
020 4
8–12 –6
–1 9 –1
D)
020 4
0–12 4
1 9 –1
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)
{(–6, 8, 12)}
B)
{(6, 5, 8)}
C)
{(6, 8, 5)}
D)
{(12, 8, –6)}
Evaluate the determinant.
71)
900
744
695
71)
A)
504
B)
–139
C)
144
D)
–144
24
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
72)
x + y + z =9
2x – 3y + 4z =7
x – 4y + 3z = – 2
72)
A)
{(z
5+34
5, 2z
5+11
5, z)}
B)
{(7z
5+34
5, 2z
5–11
5, z)}
C)
{(–7z
5+34
5, 2z
5+11
5, z)}
D)
{(–7z
5+34
5, 2z
5–11
5, z)}
73)
2x + y + 2z – 4w = 10
x + 3y + 2z – 11w = 17
3x + y + 7z – 21w = 0
73)
A)
{(–3w + 5, 2w + 6, 4w – 3, w)}
B)
{(w – 5, 8w – 4, –3w + 2, w)}
C)
{(3w + 5, 6w + 6, –4w – 3, w)}
D)
{(w + 5, 8w + 4, –3w – 2, w)}
Find values for the variables so that the matrices are equal.
74)
x + 3 y + 4
7–1
=8 1
7 z
74)
A)
x =5; y = – 1; z =8
B)
x =8; y =1; z = – 1
C)
x =5; y = – 3; z = – 1
D)
x = – 5; y =3; z =1
Find the products AB and BA to determine whether B is the multiplicative inverse of A.
75)
A =5 3
3 2 ,B =2–3
–3 5
75)
A)
B A–1
B)
B =A–1
25
Solve the problem using matrices.
76)
State University has a College of Arts & Sciences, a College of Business, and a College of
Engineering. The percentage of students in each category are given by the following matrix.
Freshman Sophomore Junior Senior
Arts & Sciences
Business
Engineering
60% 50% 40% 70%
20% 40% 30% 10%
20% 10% 30% 20%
The student population is distributed by class and age as given in the following matrix.
Female Male
Freshman
Sophomore
Junior
Senior
410 770
550 750
890 650
630 480
How many female students are in the College of Business? How many male students are in the
College of Arts & Sciences?
76)
A)
697 students; 520 students
B)
1318 students; 632 students
C)
632 students; 1433 students
D)
530 students; 697 students
Use Cramer’s rule to solve the system.
77)
3x = – 7+ 2y
4y =26 + 2x
77)
A)
{(8, 3)}
B)
{(3, –8)}
C)
{(–8, 3)}
D)
{(3, 8)}
26
Find the product AB, if possible.
78)
A =–1 3
1 4 , B =0–2 5
1–3 2
78)
A)
3 4
–7–14
113
B)
0–615
1–12 8
C)
AB is not defined.
D)
3–7 1
4–14 13
Write the augmented matrix for the system of equations.
79)
7x + 3z = 39
5y + 3z = 2
4x + 3y + 2z = 25
79)
A)
7 0 4 39
0 5 3 2
3 3 2 25
B)
7 0 3 39
0 5 3 2
4 3 2 25
C)
7 3 0 39
5 3 0 2
4 3 2 25
D)
7 0 3
0 5 3
4 3 2
Use Cramer’s rule to solve the system.
80)
5x = – 4y + 5
4x = – y – 7
80)
A)
{(–5, –3)}
B)
{(–3, –5)}
C)
{(–3, 5)}
D)
{(5, –3)}
27
Solve the problem.
81)
Determinants are used to show that three points lie on the same line (are collinear). If
x1y1 1
x2y2 1
x3y3 1
= 0,
then the points (x1, y1), (x2, y2), and (x3, y3) are collinear. If the determinant does not equal 0, then
the points are not collinear. Are the points (–1, 10), (0, –9), and (–2, 30) collinear?
81)
A)
Yes
B)
No
Write a system of linear equations in three variables, and then use matrices to solve the system.
82)
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.
82)
A)
y =6x2+20x +22
B)
y =5x2+10x +27
C)
y =7x2– 10x +30
D)
y =10x2– 30x +23
Solve the system of equations using matrices. Use Gaussian elimination with back–substitution.
83)
6x – y – 7z= – 48
4x – 5z= – 31
4y + z = 27
83)
A)
{(1, 7, 5)}
B)
{(1, 5, 7)}
C)
{(–1, 5, 2)}
D)
{(–1, 2, 5)}
28
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.
84)
3x + 8y= 48
6y= 18
84)
A)
48 8
18 0
x
y
=3
6
B)
3 8
0 6
x
y
=48
18
C)
3 8
618
x
y
=48
0
D)
6 0
3 8
x
y
=18
8
Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the
variables.
85)
9 1 0 5 –5
–1 4 1 0 4
6 0 0 7 9
0 8 0 –812
85)
A)
9x +y+z+5w = – 5
–x+4y +z+w=4
6x +y+z+7w =9
x+8y +z–8w =12
B)
9x +y+5w = – 5
x+4y +z=4
6x +7w =9
8y +8w =12
C)
9x +y+5z = – 5
–x+4y +z=4
6x +7y =9
8x –8y =12
D)
9x +y+5w = – 5
–x+4y +z=4
6x +7w =9
8y –8w =12
Find the products AB and BA to determine whether B is the multiplicative inverse of A.
86)
A =10 1
–1 0 ,B =0 1
–110
86)
A)
B =A–1
B)
B A–1
29
87)
A =
1 0 0
1 1 0
1 1 1
,B =
1 0 0
–1 1 0
0 –1 1
87)
A)
B A–1
B)
B =A–1
Find the inverse of the matrix, if possible.
88)
A =
1 2 0 0
0 1 2 0
0 0 1 –9
0 0 0 1
88)
A)
1 0 0 0
9 1 0 0
18 –2 1 0
36 –4–2 1
B)
1 0 0 0
–2 1 0 0
4–2 1 0
–18 –18 9 1
C)
1 9 18 36
0 1 –2–4
0 0 1 –2
0 0 0 1
D)
1–2 4 36
0 1 –2–18
0 0 1 9
0 0 0 1
Write a system of linear equations in three variables, and then use matrices to solve the system.
89)
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?
89)
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
30
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
90)
x + y + z + w =7
3x – 2z + 5w =11
–4x + 3y + w =4
–x – y – z – w =6
90)
A)
B)
{(3
2, 1, 1
3, –2)}
C)
{(–11, 7
19 , 6
19 , –4)}
D)
{(7
4, –1
2, 5, –1
6)}
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.
91)
Adjust the contrast by changing the black to dark grey and the light grey to white. Use matrix
addition to accomplish this.
91)
A)
1 3 1
1 3 1
3 3 3
+
0 –1 0
0 –1 0
–1–1–1
=
1 2 1
1 2 1
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
+
–1–1–1
–1–1–1
–1–1–1
=
0 2 0
0 2 0
2 2 2
31
Find the product AB, if possible.
92)
A =1–7 8
5–1–4, B =
–7
6
–1
92)
A)
AB is not defined.
B)
1–7 8
5–1–4
–7 6 –1
C)
–57
–37
D)
–57 –37
Solve the system of equations using matrices. Use Gaussian elimination with back–substitution.
93)
x – y + 3z= 2
3x + z =2
x + 4y + z =18
93)
A)
{(0, 2, 4)}
B)
{(0, 4, 2)}
C)
{(2, 0, 4)}
D)
{(2, 4, 0)}
Solve the problem.
94)
The equation of a line passing through two distinct points (x1, y1) and (x2, y2) is given by
x y 1
x2y2 1
x3y3 1
= 0. Use the determinant to write an equation for the line passing through (9, 9)
and (–7, 3). Express the line’s equation in standard form.
94)
A)
6x – 16y + 90 = 0
B)
3x + 9y – 63 = 0
C)
–6x + 16y + 90 = 0
D)
9x – 7y + 27 = 0
32
Find the inverse of the matrix, if possible.
95)
A =–4 6
–1–6
95)
A)
No inverse
B)
–4–6
–1–6
C)
–1
4
1
6
– 1 1
6
D)
4 6
–1–6
Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the
variables. Then use back–substitution to find the solution.
96)
1 1 –1 1 –8
0 1 –9 9 0
0 0 1 4 18
0 0 0 1 3
96)
A)
{(–10, –1, 13, 2)}
B)
{(–32, 27, 6, 3)}
C)
{(3, 6, 27, –32)}
D)
{(–8, 0, 18, 3)}
Use Cramer’s rule to determine if the system is inconsistent system or contains dependent equations.
97)
4x – y + 2z = 1
3x + 5y – z = 0
–6x – 10y + 2z = 0
97)
A)
system contains dependent equations
B)
system is inconsistent
Evaluate the determinant.
98)
2 3
1–1
98)
A)
7
B)
1
C)
–5
D)
5
33
Solve the problem.
99)
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.
99)
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
Use Cramer’s rule to solve the system.
100)
–4x – 4y – 3z = – 22
6x + 2y – 3z =14
–9x – 5y + 8z = – 16
100)
A)
{(3, –1, –2)}
B)
{(4, –1, 2)}
C)
{(3, 1, 2)}
D)
{(1, 2, 1)}
101)
9x + 5y – z =90
x + 9y + 9z =168
7x + y + z =60
101)
A)
{(6, 9, 9)}
B)
{(6, –9, –9)}
C)
{(9, 9, 9)}
D)
{(7, 7, 9)}
34
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
102)
5x – y + z = 8
7x + y + z = 6
102)
A)
{(–z + 3, 4z + 7, z)}
B)
C)
{(–1
6z +7
6, 1
6z –13
6, z)}
D)
{(1
6z +7
6, 1
6z, z)}
Find the inverse of the matrix, if possible.
103)
A =2–6
–2 0
103)
A)
–1
6–1
6
0–1
2
B)
–1
6–1
2
–1
60
C)
01
2
1
6–1
6
D)
0–1
2
–1
6–1
6
Solve the system of equations using matrices. Use Gaussian elimination with back–substitution.
104)
x + y + z – w =6
2x – y + 3z + 4w = – 4
4x + 2y – z – w = – 13
–x – 2y + 4z + 3w =12
104)
A)
{(–4, 3, 5, –2)}
B)
{(–1
4, 1
3, 1
5, –1
2)}
C)
{(1
4, –1
3, –1
5, 1
2)}
D)
{(4, –3, –5, 2)}
35
Write the augmented matrix for the system of equations.
105)
8x +9y –4z + w =3
10y + z =12
x – y –12z =8
2x –2y +6z = – 4
105)
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 0 1 9
910 –1–2
–4 1 –12 6
1 0 0 0
312 8–4
D)
8 9 –4 1 3
010 1 0 12
1–1–12 0 8
2–2 6 0 –4
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
106)
x – y + z – w = 10
–2x + 3y + 5w = – 28
x + 2y + 8z + 3w = – 10
x – 4y – 6z – 5w = 30
106)
A)
{(–17w – 10, –13w – 16, 5w + 4, w)}
B)
C)
{(24, 10, –6, –2)}
D)
{(3w – 2, –8w + 3, 4w + 9, w)}
Find the inverse of the matrix, if possible.
107)
A =
1 0 0
–9 1 0
0–7 1
107)
A)
1 0 0
–7–1 0
–63 –9 1
B)
1 0 0
–9 1 0
0 0 –7
C)
1–7–63
0 1 1
0 0 1
D)
100
910
63 7 1
36
Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the
variables.
108)
9 5 9 –2
8 0 8 4
7 4 0 2
108)
A)
9x + 5y + 9z = – 2
8x + y + 8z = 4
7x + 4y + z = 2
B)
9x + 5y + 9z = – 2
8x + 8z = 4
7x + 4y = 2
C)
9x + 5y + 9z = – 2
8x + 8z = 4
7x + 4z = 2
D)
9x – 5y + 9z = – 2
8x + 8z = – 4
7x + 4y = – 2
Solve the problem.
109)
The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is
Area = ± 1
2
x1y1 1
x2y2 1
x3y3 1
,
where the symbol ± indicates that the appropriate sign should be chosen to yield a positive area.
Use this formula to find the area of a triangle whose vertices are (10, 4), (3, –5), and (–7, –4).
109)
A)
3
B)
3
2
C)
97
D)
97
2
Find the products AB and BA to determine whether B is the multiplicative inverse of A.
110)
A =
2–1 0
–1 1 –2
1 0 –1
,B =
1–1 2
–3–2 4
–1 1 1
110)
A)
B =A–1
B)
B A–1
37
Use Cramer’s rule to solve the system.
111)
5x + 4y – z =28
x – 3y + 4z =32
2x + y + z =22
111)
A)
{(6, 1, 9)}
B)
{(5, 3, 9)}
C)
{(3, 9, 3)}
D)
{(5, –3, –9)}
Solve the problem.
112)
Let B = [–1 3 7 –3]. Find –2B.
112)
A)
–2 6 14 –6
B)
2 3 7 –3
C)
–3 1 5 –5
D)
2–6–14 6
Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the
variables. Then use back–substitution to find the solution.
113)
1 7 –9–2
0 1 2 –5
0 0 1 3
113)
A)
{(–2, –5, 3)}
B)
{(–1, –8, 2)}
C)
{(102, –11, 3)}
D)
{(–22, 1, 3)}
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
114)
x + 3y + 2z =11
4y + 9z = – 12
x + 7y + 11z = – 1
114)
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)}
38