Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use Cramer’s rule to solve the system.
1)
5x + 4y=- 8
2x + y = – 5
1)
A)
{(–3, –4)}
B)
{(–4, –3)}
C)
{(–4, 3)}
D)
{(3, –4)}
Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the
variables.
2)
9 5 9 –2
8 0 8 4
7 4 0 2
2)
A)
9x + 5y + 9z = – 2
8x + 8z = 4
7x + 4y = 2
B)
9x + 5y + 9z = – 2
8x + 8z = 4
7x + 4z = 2
C)
9x + 5y + 9z = – 2
8x + y + 8z = 4
7x + 4y + z = 2
D)
9x – 5y + 9z = – 2
8x + 8z = – 4
7x + 4y = – 2
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.
3)
–2x + 9z= 5
3y + 8z = 20
5x + 8y + 7z = 49
3)
A)
–2 0
0 3
5 8
x
y
z
=
9
8
7
B)
–2 0 5
0 3 8
9 8 7
x
y
z
=
5
20
49
C)
–2 9 0
3 8 0
5 8 7
x
y
z
=
5
20
49
D)
–2 0 9
0 3 8
5 8 7
x
y
z
=
5
20
49
1
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
4)
4x – y + 3z =12
x + 4y + 6z = – 32
5x + 3y + 9z =20
4)
A)
{(2, –7, –1)}
B)
{(8, –7, –2)}
C)
{(–8, –7, 9)}
D)
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.
5)
–2x + 3y = 10
5x – 2y = 19
5)
A)
10 3
19 5
x
y
=–2
–2
B)
–2 5
3–2
x
y
=10
19
C)
–2 3
–2 5
x
y
=19
10
D)
–2 3
5–2
x
y
=10
19
Solve the system of equations using matrices. Use Gaussian elimination with back–substitution.
6)
6x – y – 7z= – 48
4x – 5z= – 31
4y + z = 27
6)
A)
{(–1, 5, 2)}
B)
{(1, 7, 5)}
C)
{(–1, 2, 5)}
D)
{(1, 5, 7)}
Write the augmented matrix for the system of equations.
7)
7x + 3z = 39
5y + 3z = 2
4x + 3y + 2z = 25
7)
A)
7 0 3 39
0 5 3 2
4 3 2 25
B)
7 3 0 39
5 3 0 2
4 3 2 25
C)
7 0 3
0 5 3
4 3 2
D)
7 0 4 39
0 5 3 2
3 3 2 25
2
Solve the problem.
8)
Let A =–1 0
3 1 and B =–1 3
3 1 . Find A – B.
8)
A)
–3
B)
–2 3
6 2
C)
0 3
0 0
D)
0–3
0 0
Find the products AB and BA to determine whether B is the multiplicative inverse of A.
9)
A =
1 0 0
1 1 0
1 1 1
,B =
1 0 0
–1 1 0
0 –1 1
9)
A)
B A–1
B)
B =A–1
Use Cramer’s rule to determine if the system is inconsistent system or contains dependent equations.
10)
8x + y =25
8x + y =49
10)
A)
system is inconsistent
B)
system contains dependent equations
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
11)
x + y + z =9
2x – 3y + 4z =7
x – 4y + 3z = – 2
11)
A)
{(–7z
5+34
5, 2z
5+11
5, z)}
B)
{(7z
5+34
5, 2z
5–11
5, z)}
C)
{(z
5+34
5, 2z
5+11
5, z)}
D)
{(–7z
5+34
5, 2z
5–11
5, z)}
3
Write a system of linear equations in three variables, and then use matrices to solve the system.
12)
There were approximately 100,000 vehicles sold at a particular dealership last year. The dealer
tracks sales by age group for marketing purposes. The percentage of 36– to 59–year–old buyers
and the percentage of buyers 60 and older combined exceeds the percentage of buyers 35 and
younger by 38%. If the percentage of buyers in the oldest group is doubled, it is 36% less than the
percentage of users in the middle group. Find the percentage of buyers in each of the three age
groups.
12)
A)
11% 35 and younger; 58% 36–59 year olds; 31% 60 and older
B)
31% 35 and younger; 58% 36–59 year olds; 11% 60 and older
C)
33% 35 and younger; 55% 36–59 year olds; 12% 60 and older
D)
25% 35 and younger; 60% 36–59 year olds; 15% 60 and older
Find the inverse of the matrix, if possible.
13)
A =–4 6
–1–6
13)
A)
No inverse
B)
4 6
–1–6
C)
–1
4
1
6
– 1 1
6
D)
–4–6
–1–6
Solve the matrix equation for X.
14)
Let A =1 4
3 1 and B =3–5
1–5;X + A = B
14)
A)
X =–2–6
2–9
B)
X =–9 2
–6–2
C)
X =2–9
–2–6
D)
X =–6–2
–9 2
4
Evaluate the determinant.
15)
0 7 9 8
0 4 9 4
4 7 2 4
5 1 6 6
15)
A)
1280
B)
6
C)
–36
D)
20
Find the product AB, if possible.
16)
A =3–2 1
0 4 –1, B =5 0
–2 2
16)
A)
15 –10 5
–612 –4
B)
15 –6
–10 12
5–4
C)
AB is not defined.
D)
15 0
0 8
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.
17)
1 1 –1 1 –8
0 1 –9 9 0
0 0 1 4 18
0 0 0 1 3
17)
A)
{(–32, 27, 6, 3)}
B)
{(–10, –1, 13, 2)}
C)
{(3, 6, 27, –32)}
D)
{(–8, 0, 18, 3)}
Solve the problem.
18)
Let A =
3
–6
–4
and B =
–5
6
5
. Find A + B.
18)
A)
2
1
2
B)
–2
0
1
C)
[–2 0 1]
D)
3–5
–6 6
–4 5
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
19)
2x + y + 2z – 4w = 10
x + 3y + 2z – 11w = 17
3x + y + 7z – 21w = 0
19)
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)}
Solve the problem.
20)
Let A =
1
–3
2
and B =
–1
3
–2
. Find A – 2B.
20)
A)
3
–9
6
B)
3
–6
4
C)
–3
9
–6
D)
–1
3
–2
Use Cramer’s rule to solve the system.
21)
5x = – 4y + 5
4x = – y – 7
21)
A)
{(5, –3)}
B)
{(–3, –5)}
C)
{(–5, –3)}
D)
{(–3, 5)}
6
Find the inverse of the matrix, if possible.
22)
A =0 6
–6 4
22)
A)
1
9
1
6
–1
60
B)
0–1
6
1
6
1
9
C)
1
60
1
9–1
6
D)
1
9–1
6
1
60
Evaluate the determinant.
23)
3 4
6 5
23)
A)
39
B)
–18
C)
9
D)
–9
Solve the matrix equation for X.
24)
Let A =4 2
3 5 and B =3–4
4–5;X + A = B
24)
A)
X =–6–1
–10 1
B)
X =–1–6
1–10
C)
X =1–10
–1–6
D)
X =–10 –1
–6 1
Write the augmented matrix for the system of equations.
25)
x –9y + z =15
y +6z =17
z =11
25)
A)
1–9 1 15
0 1 6 17
0 0 1 11
B)
1–9 1 15
1 1 6 17
1 1 1 11
C)
1 9 1 15
0 1 6 17
0 0 1 11
D)
0–9 0 15
0 0 6 17
0 0 0 11
Solve the problem.
26)
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?
26)
A)
Yes
B)
No
Write the matrix equation as a system of linear equations without matrices.
27)
20 12
13 18
x
y
=2
20
27)
A)
12x + 20y = 2
13x + 18y = 20
B)
20x + 12y = 2
18x + 13y = 20
C)
20x + 12y = 2
13x + 18y = 20
D)
20x + 12y = – 2
13x + 18y = – 20
Find values for the variables so that the matrices are equal.
28)
x + 3 y + 4
7–1
=8 1
7 z
28)
A)
x =5; y = – 3; z = – 1
B)
x =8; y =1; z = – 1
C)
x = – 5; y =3; z =1
D)
x =5; y = – 1; z =8
Solve the system of equations using matrices. Use Gauss–Jordan elimination.
29)
3x + 5y + 2w = – 12
2x + 6z – w = – 5
–2y + 3z – 3w =– 3
–x + 2y + 4z + w = – 2
29)
A)
{(–1, 3, 0, –3)}
B)
{(–1, –3, 0, 3)}
C)
{(1, 3, 0, –3)}
D)
{(1, –3, 0, 3)}
Find the inverse of the matrix, if possible.
30)
A =
1 0 0
–9 1 0
0–7 1
30)
A)
100
910
63 7 1
B)
1 0 0
–7–1 0
–63 –9 1
C)
1 0 0
–9 1 0
0 0 –7
D)
1–7–63
0 1 1
0 0 1
Find the products AB and BA to determine whether B is the multiplicative inverse of A.
31)
A =–5–1
6 0 ,B =
01
6
–15
6
31)
A)
B =A–1
B)
B A–1
Solve the problem.
32)
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.
32)
A)
3x + 9y – 63 = 0
B)
–6x + 16y + 90 = 0
C)
9x – 7y + 27 = 0
D)
6x – 16y + 90 = 0
Find the products AB and BA to determine whether B is the multiplicative inverse of A.
33)
A =
2 0 1 1
1 1 –1–1
–1–2 1 0
0 0 1 1
,B =
2 0 0 –2
–2 1 0 3
–2 2 1 5
2 –2–1–3
33)
A)
B A–1
B)
B =A–1
Solve the problem.
34)
Let A =3 3
2 5 and B =0 4
–1 6 . Find 4A + B.
34)
A)
12 7
711
B)
12 16
726
C)
12 16
111
D)
12 28
444
Find the inverse of the matrix, if possible.
35)
A =
100
–110
111
35)
A)
1–1 1
0 1 –1
0 0 1
B)
–1 0 0
–1–1 0
–1–1–1
C)
1 0 0
1 1 0
–2–1 1
D)
1 1 1
0 1 1
0 0 1
Find the product AB, if possible.
36)
A =3–2 1
0 4 –3, B =5 0
–2 2
36)
A)
AB is not defined.
B)
15 0
0 8
C)
15 –6
–10 12
5–8
D)
15 –10 5
–612 –8
Write the matrix equation as a system of linear equations without matrices.
37)
6 2 9
5 0 2
8 8 0
x
y
z
=
–2
4
2
37)
A)
6x + 2y + 9z= – 2
5x + 2z=4
8x + 8z=2
B)
6x + 2y + 9z= – 2
5x + 2z=4
8x + 8y=2
C)
6x + 2y + 9z= – 2
5x + 2y=4
8x + 8z=2
D)
6x – 2y + 9z= – 2
5x + 2z= – 4
8x + 8y= – 2
Solve the system using the inverse that is given for the coefficient matrix.
38)
x+2y +3z =4
x+y+z=7
x–2z =11
The inverse of
1 2 3
1 1 1
1 0 –2
is
–2 4 –1
3 –5 2
–1 2 –1
.
38)
A)
{(47, 69, 29)}
B)
{(4, 0, 0)}
C)
{(9, –1, –1)}
D)
{(2, 3, –1)}
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.
39)
Adjust the contrast by leaving the black alone and changing the light grey to dark grey. Use matrix
addition to accomplish this.
39)
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
3 2 2
B)
131
131
333
+
1 –1 1
1 –1 1
–1–1–1
=
1 2 1
1 2 1
3 2 2
C)
1 3 1
1 3 1
3 3 3
+
1 0 1
1 0 1
0 0 0
=
2 3 2
2 3 2
3 3 3
D)
131
131
333
+
1 1 1
1 1 1
1 1 1
=
2 3 2
2 3 2
3 3 3
Solve the problem.
40)
Let B = [–1 3 7 –3]. Find –2B.
40)
A)
2 3 7 –3
B)
2–6–14 6
C)
–3 1 5 –5
D)
–2 6 14 –6
Give the order of the matrix, and identify the given element of the matrix.
41)
615 –4 3 –10
–11 –e–5 2
112 –812 0
1
3115 11 8
; a34
41)
A)
20; 0
B)
4 × 4;–8
C)
5 × 4;15
D)
4 × 5;12
Find the products AB and BA to determine whether B is the multiplicative inverse of A.
42)
A =–2 4
4–4,B =
1
2
1
4
1
2
1
4
42)
A)
B =A–1
B)
B A–1
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
43)
3x + y + z – 2w = 10
2x + 3y + 3z + w = – 5
2x + y + 4z + 11w = 11
43)
A)
{(2w + 3, 6w – 7, –10w + 8, w)}
B)
{(6, –4, –2, 1)}
C)
{(7, –1, –6, 2)}
D)
{(w + 5, 3w – 7, –4w + 2, w)}
13
Find the products AB and BA to determine whether B is the multiplicative inverse of A.
44)
A =
2–1 0
–1 1 –2
1 0 –1
,B =
1–1 2
–3–2 4
–1 1 1
44)
A)
B =A–1
B)
B A–1
Encode or decode the given message, as requested, numbering the letters of the alphabet 1 through 26 in their usual order.
45)
Use the coding matrix A = 1 –4
–2 9 and its inverse A–1=9 4
2 1 to decode the cryptogram –7–8
16 21 .
45)
A)
ABLE
B)
ARMS
C)
ACTS
D)
ALAS
Find values for the variables so that the matrices are equal.
46)
x
7
=4
y
46)
A)
x =4; y =7
B)
x =7; y =7
C)
x =7; y =4
D)
x =4; y =4
Perform the matrix row operation (or operations) and write the new matrix.
47)
15 –35 –10 –50
113 –3 0
2–7 4 21
1
5R1
47)
A)
3–7–2–50
113 –3 0
2–7 4 21
B)
3–7–2–10
113 –3 0
2–7 4 21
C)
15 –35 –10 –50
1
5
13
5–3
50
2–7 4 21
D)
3–7–2–10
1
5
13
5–3
50
2
5–7
5
4
5
21
5
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
48)
x + 8y + 8z =8
7x + 7y + z =1
8x + 15y + 9z = – 9
48)
A)
B)
{(–1, 0, 1)}
C)
{(0, 0, 1)}
D)
{(1, –1, 1)}
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.
49)
3x + 8y= 48
6y= 18
49)
A)
6 0
3 8
x
y
=18
8
B)
3 8
618
x
y
=48
0
C)
48 8
18 0
x
y
=3
6
D)
3 8
0 6
x
y
=48
18
15
Solve the system of equations using matrices. Use Gauss–Jordan elimination.
50)
x + y – z + w = – 5
3x – y + 3z – 2w = 7
–2x + 2y + z – w =16
–x – 2y – 3z + 3w = – 22
50)
A)
{(–2, –3, 5, 1
2)}
B)
{(–2, 3, 4, –2)}
C)
{(2, –3, –4, –2)}
D)
{(1
2, –1
3, –1
4, –1
2)}
Encode or decode the given message, as requested, numbering the letters of the alphabet 1 through 26 in their usual order.
51)
Use the coding matrix A =
1 1 1
–1 1 2
1 2 3
and its inverse A–1=
–1–1 1
5 2 –3
–3–1 2
to decode the cryptogram
37 16 35
38 20 4
82 40 60
.
51)
A)
STAY_CALM
B)
HELP_THEM
C)
GOOD_LUCK
D)
LOOK_DOWN
Solve the problem using matrices.
52)
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?
52)
A)
1318 students; 632 students
B)
530 students; 697 students
C)
697 students; 520 students
D)
632 students; 1433 students
Perform the matrix row operation (or operations) and write the new matrix.
53)
3–4 1 2
–5 0 2 –2
–1 4 –3–1
–3R1+R2
53)
A)
–14 12 –1–8
–5 0 2 –2
–1 4 –3–1
B)
3–4 1 2
4–12 5 4
–1 4 –3–1
C)
18 –4–5 8
–5 0 2 –2
–1 4 –3–1
D)
3–4 1 2
–14 12 –1–8
–1 4 –3–1
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.
54)
13
219
2
0 1 1
2–2
0 0 1 6
54)
A)
{(6, –5, 6)}
B)
{(1, –7
2, 5)}
C)
{(9
2, –2, 6)}
D)
{(12, –5, 6)}
Find the inverse of the matrix, if possible.
55)
A =–6–2
0 5
55)
A)
01
5
–1
6–1
15
B)
–1
6
1
15
01
5
C)
1
5–1
15
0–1
6
D)
–1
6–1
15
01
5
Solve the system of equations using matrices. Use Gaussian elimination with back–substitution.
56)
x + y + z = – 2
x – y + 5z=26
5x + y + z = – 14
56)
A)
{(5, –3, –4)}
B)
{(–3, 5, –4)}
C)
{(–3, –4, 5)}
D)
{(5, –4, –3)}
18
Solve the matrix equation for X.
57)
Let A =
3 5
0 1
7–9
and B =
6–6
–3 5
0 6
;B – X = 3A
57)
A)
X =
–3–21
–3 2
–21 33
B)
X =
–3–21
3 2
–21 33
C)
X =
15 9
3 8
21 –21
D)
X =
15 9
3 8
7–21
Perform the matrix row operation (or operations) and write the new matrix.
58)
1 1 –1 1 3
0–4 3 –3 0
4 0 –3–1 3
–3 2 0 3 –2
–4R1+R3
3R1+R4
58)
A)
1 1 –1 1 3
0–4 3 –3 0
0–4 1 –5–1
0 5 –3 6 4
B)
1 1 –1 1 3
0–4 3 –3 0
0–4 1 –5–9
0 5 –3 6 7
C)
1 1 –1 1 3
0–4 3 –3 0
0–4 1 –5–9
–3 2 0 3 –2
D)
1 1 –1 1 3
0–4 3 –3 0
8 4 –7 3 15
0 5 –3 6 7
Solve the problem.
59)
Let A =–9 1
2 5 and B =6 2
1 1 . Find A + B.
59)
A)
–3 3
3 6
B)
–3–7
–8–8
C)
3 4
2 6
D)
9
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.
60)
1 7 –9–2
0 1 2 –5
0 0 1 3
60)
A)
{(–2, –5, 3)}
B)
{(102, –11, 3)}
C)
{(–22, 1, 3)}
D)
{(–1, –8, 2)}
Solve the problem.
61)
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 (–5, 5), (0, –6), and (–15, 27) collinear?
61)
A)
Yes
B)
No
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
62)
x + y + z = 9
2x – 3y + 4z = 7
62)
A)
{(3
5z +16
5, –8
5z +29
5, z)}
B)
{(27
5, 13
5, 1)}
C)
{(–7
5z +34
5, 2
5z +11
5, z)}
D)
Use Cramer’s rule to solve the system.
63)
2x + 3y + 2z =43
–2y + 4z =6
3x + 2z =28
63)
A)
{(7, 5, 5)}
B)
{(6, –7, –5)}
C)
{(6, 7, 5)}
D)
{(7, 5, 7)}
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