Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use Cramer’s rule to determine if the system is inconsistent system or contains dependent equations.
1)
–2x – 7y= – 47
–4x – 14y= – 49
1)
A)
system is inconsistent
B)
system contains dependent equations
2)
x + z = 1
2x – 2y = – 2
y + z = 4
2)
A)
system is inconsistent
B)
system contains dependent equations
Find the inverse of the matrix, if possible.
3)
A =
100
–110
111
3)
A)
1–1 1
0 1 –1
0 0 1
B)
1 0 0
1 1 0
–2–1 1
C)
–1 0 0
–1–1 0
–1–1–1
D)
1 1 1
0 1 1
0 0 1
1
Solve the problem using matrices.
4)
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?
4)
A)
t + 2 cars/min between I2 and I1; t – 3 cars/min between I1 and I3
B)
t + 1 cars/min between I2 and I1; t + 4 cars/min between I1 and I3
C)
t – 2 cars/min between I2 and I1; t + 1 cars/min between I1 and I3
D)
t + 8 cars/min between I2 and I1; t + 3 cars/min between I1 and I3
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
5)
x + 8y + 8z =8
7x + 7y + z =1
8x + 15y + 9z = – 9
5)
A)
B)
{(1, –1, 1)}
C)
{(0, 0, 1)}
D)
{(–1, 0, 1)}
2
Write the augmented matrix for the system of equations.
6)
7x + 7y + 8z= 25
9x + 3y + 3z= 15
3x + 3y – 2z= – 11
6)
A)
7 7 8
9 3 3
3 3 –2
B)
25 877
15 339
–11 –233
C)
7 7 8 25
9 3 3 15
3 3 –2–11
D)
7 9 3 25
7 3 3 15
8 3 –2–11
Encode or decode the given message, as requested, numbering the letters of the alphabet 1 through 26 in their usual order.
7)
Use the coding matrix A =–1–3
2 5 to encode the message CARE.
7)
A)
18 105
–7–4
B)
–6–33
11 61
C)
–57 –16
96 27
D)
–1–8
–4–29
Solve the system of equations using matrices. Use Gaussian elimination with back–substitution.
8)
x + y + z = – 2
x – y + 5z=26
5x + y + z = – 14
8)
A)
{(–3, 5, –4)}
B)
{(–3, –4, 5)}
C)
{(5, –4, –3)}
D)
{(5, –3, –4)}
3
Solve the problem using matrices.
9)
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?
9)
A)
7 oz of Food A; 7 oz of Food B; 1 oz of Food C
B)
4 oz of Food A; 6 oz of Food B; 2 oz of Food C
C)
No possible combination of these foods
D)
3 oz of Food A; 5 oz of Food B; 1 oz of Food C
Give the order of the matrix, and identify the given element of the matrix.
10)
6 8 7 –7
–9–1 4 0 ; a12
10)
A)
4 × 2; –9
B)
2 × 4; 8
C)
2 × 4;–9
D)
4 × 2; 8
Write the matrix equation as a system of linear equations without matrices.
11)
6 2 9
5 0 2
8 8 0
x
y
z
=
–2
4
2
11)
A)
6x + 2y + 9z= – 2
5x + 2z=4
8x + 8y=2
B)
6x + 2y + 9z= – 2
5x + 2y=4
8x + 8z=2
C)
6x + 2y + 9z= – 2
5x + 2z=4
8x + 8z=2
D)
6x – 2y + 9z= – 2
5x + 2z= – 4
8x + 8y= – 2
4
Use Cramer’s rule to solve the system.
12)
5x + 4y=- 8
2x + y = – 5
12)
A)
{(3, –4)}
B)
{(–3, –4)}
C)
{(–4, –3)}
D)
{(–4, 3)}
13)
11x +3y=1
5x +2y =4
13)
A)
{(39
7, –10
7)}
B)
{(–10
7, 39
7)}
C)
{(–7
10 , 7
39 )}
D)
{(14
37 , 49
37 )}
Solve the system of equations using matrices. Use Gauss–Jordan elimination.
14)
x = 4 – y – z
x – y + 2z= – 6
2x + y = 9 – z
14)
A)
{(–4, 5, 3)}
B)
{(3, –4, 5)}
C)
{(5, 3, –4)}
D)
{(–4, 3, 5)}
Write a system of linear equations in three variables, and then use matrices to solve the system.
15)
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.
15)
A)
33% 35 and younger; 55% 36–59 year olds; 12% 60 and older
B)
11% 35 and younger; 58% 36–59 year olds; 31% 60 and older
C)
31% 35 and younger; 58% 36–59 year olds; 11% 60 and older
D)
25% 35 and younger; 60% 36–59 year olds; 15% 60 and older
5
Solve the system of equations using matrices. Use Gauss–Jordan elimination.
16)
5x + 8y–z=36
x– 2y+ 7z=18
3x +y+z=21
16)
A)
{(6, 2, 1)}
B)
{(6, 1, 2)}
C)
{(12, 1, –6)}
D)
{(–6, 1, 12)}
Find the products AB and BA to determine whether B is the multiplicative inverse of A.
17)
A =–5 1
–7 1 ,B =
1
2–1
2
7
2–5
2
17)
A)
B A–1
B)
B =A–1
Solve the problem.
18)
Let A =–1 0
3 1 and B =–1 3
3 1 . Find A – B.
18)
A)
0–3
0 0
B)
–2 3
6 2
C)
–3
D)
0 3
0 0
Evaluate the determinant.
19)
1
5
1
6
–3
8
8
7
19)
A)
93
560
B)
163
560
C)
11
1680
D)
–223
840
6
Encode or decode the given message, as requested, numbering the letters of the alphabet 1 through 26 in their usual order.
20)
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
.
20)
A)
STAY_CALM
B)
LOOK_DOWN
C)
HELP_THEM
D)
GOOD_LUCK
Use Cramer’s rule to solve the system.
21)
–2x + 3y= – 5
4x + 3y=19
21)
A)
{(4, 1)}
B)
{(–1, 4)}
C)
{(1, 4)}
D)
{(4, –1)}
Solve the problem.
22)
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?
22)
A)
Yes
B)
No
7
Find the product AB, if possible.
23)
A =6–2–8
–4 1 8 , B =
–2
–2
6
23)
A)
–56 54
B)
AB is not defined.
C)
–56
54
D)
6–2–8
–4 1 8
–2–2 6
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.
24)
–2x + 9z= 5
3y + 8z = 20
5x + 8y + 7z = 49
24)
A)
–2 9 0
3 8 0
5 8 7
x
y
z
=
5
20
49
B)
–2 0
0 3
5 8
x
y
z
=
9
8
7
C)
–2 0 5
0 3 8
9 8 7
x
y
z
=
5
20
49
D)
–2 0 9
0 3 8
5 8 7
x
y
z
=
5
20
49
Use Cramer’s rule to solve the system.
25)
4x – 6z =4
–3x + 2y – 5z = – 20
6x – 2y =22
25)
A)
{(1, 2, 1)}
B)
{(4, –1, –2)}
C)
{(4, 1, 2)}
D)
{(5, –1, 2)}
8
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.
26)
Adjust the contrast by changing the black to light grey and the light grey to black. Use matrix
addition to accomplish this.
26)
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
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
27)
5x + 2y + z = – 11
2x – 3y – z =17
7x – y =12
27)
A)
{(0, –6, 1)}
B)
{(1, –5, 0)}
C)
{(–2, 0, –1)}
D)
9
Solve the matrix equation for X.
28)
Let A =
–6 8
5–3
4–2
and B =
9 2
–8 5
–1 8
;X – B = A
28)
A)
3–3
–3 2
3 6
B)
310
–3 2
3 6
C)
–15 6
13 –8
5–6
D)
310
3–3
3–6
Perform the matrix row operation (or operations) and write the new matrix.
29)
1 1 –1 1 3
0–4 3 –3 0
4 0 –3–1 3
–3 2 0 3 –2
–4R1+R3
3R1+R4
29)
A)
1 1 –1 1 3
0–4 3 –3 0
8 4 –7 3 15
0 5 –3 6 7
B)
1 1 –1 1 3
0–4 3 –3 0
0–4 1 –5–9
–3 2 0 3 –2
C)
1 1 –1 1 3
0–4 3 –3 0
0–4 1 –5–1
0 5 –3 6 4
D)
1 1 –1 1 3
0–4 3 –3 0
0–4 1 –5–9
0 5 –3 6 7
10
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.
30)
13
219
2
0 1 1
2–2
0 0 1 6
30)
A)
{(6, –5, 6)}
B)
{(1, –7
2, 5)}
C)
{(12, –5, 6)}
D)
{(9
2, –2, 6)}
Solve the problem.
31)
Let A =
3
–6
–4
and B =
–5
6
5
. Find A + B.
31)
A)
[–2 0 1]
B)
2
1
C)
–2
0
D)
3–5
–6 6
Find the product AB, if possible.
32)
A =1 3 –1
3 0 5 , B =
3 0
–1 1
0 5
32)
A)
3–3 0
0 0 25
B)
0–2
925
C)
–2 0
25 9
D)
AB is not defined.
11
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
Find the product AB, if possible.
34)
A =3–2 1
0 4 –1, B =5 0
–2 2
34)
A)
AB is not defined.
B)
15 –6
–10 12
5–4
C)
15 0
0 8
D)
15 –10 5
–612 –4
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.
35)
3x + 5y + 7z = 53
9x + 8y + 3z = 19
–2x – 2y – 2z = – 14
35)
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
12
Solve the problem.
36)
Let A =3 3
2 5 and B =0 4
–1 6 . Find 4A + B.
36)
A)
12 16
726
B)
12 16
111
C)
12 28
444
D)
12 7
711
Write the matrix equation as a system of linear equations without matrices.
37)
20 12
13 18
x
y
=2
20
37)
A)
20x + 12y = 2
18x + 13y = 20
B)
12x + 20y = 2
13x + 18y = 20
C)
20x + 12y = – 2
13x + 18y = – 20
D)
20x + 12y = 2
13x + 18y = 20
Solve the system of equations using matrices. Use Gauss–Jordan elimination.
38)
x + y – z + w = – 5
3x – y + 3z – 2w = 7
–2x + 2y + z – w =16
–x – 2y – 3z + 3w = – 22
38)
A)
{(–2, –3, 5, 1
2)}
B)
{(2, –3, –4, –2)}
C)
{(1
2, –1
3, –1
4, –1
2)}
D)
{(–2, 3, 4, –2)}
Use Cramer’s rule to solve the system.
39)
2x + 3y + 2z =43
–2y + 4z =6
3x + 2z =28
39)
A)
{(7, 5, 5)}
B)
{(6, 7, 5)}
C)
{(6, –7, –5)}
D)
{(7, 5, 7)}
13
Solve the problem.
40)
Let A =
2–8
–8 1
6–7
and B =
–2–8
–3–1
–4–8
. Find A + B.
40)
A)
0–16
–11 0
2–15
B)
0 1
–11 0
2–15
C)
4 0
–5 2
10 2
D)
0–16
11 1
215
41)
Let A = [–2 2] and B = [1 0]. Find 2A + 3B.
41)
A)
–4 4
B)
–3 4
C)
–1 4
D)
1 2
Solve the system of equations using matrices. Use Gauss–Jordan elimination.
42)
3x + 5y + 2w = – 12
2x + 6z – w = – 5
–2y + 3z – 3w =- 3
–x + 2y + 4z + w = – 2
42)
A)
{(1, –3, 0, 3)}
B)
{(–1, –3, 0, 3)}
C)
{(1, 3, 0, –3)}
D)
{(–1, 3, 0, –3)}
Evaluate the determinant.
43)
–4 1 2
–1 0 –2
5 0 1
43)
A)
9
B)
11
C)
–11
D)
–9
14
Solve the problem using matrices.
44)
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.
44)
A)
System 1: 70.5; System 2: 104.5
B)
System 1: 77.2; System 2: 76
C)
System 1: 44.2; System 2: 53
D)
System 1: 88.5; System 2: 87.5
Use Cramer’s rule to solve the system.
45)
2x + 3y=34
2x –2y =4
45)
A)
{(–6, 8)}
B)
{(8, 6)}
C)
{(–8, –6)}
D)
{(6, 8)}
15
Find the product AB, if possible.
46)
A =
2 –7–9
–6–9–1
1 4 1
, B =
–6–5–3
5 4 4
5 2 7
46)
A)
–92 –14 19
–56 –8 13
–97 –25 20
B)
–12 35 27
–30 –36 –4
587
C)
2–7–9
–6–9–1
1 4 1
–6–5–3
5 4 4
5 2 7
D)
–92 –56 –97
–14 –8–25
19 13 20
Find the inverse of the matrix, if possible.
47)
A =3–2
2 6
47)
A)
3
11 –1
11
1
11
3
22
B)
3
22
1
11
–1
11
3
11
C)
3
11
1
11
–1
11
3
22
D)
–1
11
3
22
3
11
1
11
16
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
48)
x + y + z = 7
x – y + 2z = 7
48)
A)
{(8, –3, 2)}
B)
{(4, 1, 2)}
C)
{(–3z + 14, 2z – 7, z)}
D)
{(–3
2z + 7, 1
2z, z)}
Use Cramer’s rule to determine if the system is inconsistent system or contains dependent equations.
49)
8x + y =25
8x + y =49
49)
A)
system contains dependent equations
B)
system is inconsistent
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
50)
x + y + z + w = 8
3x + 2y + z + 4w = 21
4x + 4y + 5z + 8w = 30
2x + 3y + 6z + 9w = 15
50)
A)
{(–3, 16, –6, 1)}
B)
C)
{(5w + 11, –3w – 7, –3w + 4, w)}
D)
{(–6w + 3, 9w + 7, –4w – 2, w)}
Find the inverse of the matrix, if possible.
51)
A =0 6
–6 4
51)
A)
1
60
1
9–1
6
B)
1
9–1
6
1
60
C)
1
9
1
6
–1
60
D)
0–1
6
1
6
1
9
17
Evaluate the determinant.
52)
5 3 4 5
1 9 6 2
4 7 0 5
6 9 3 9
52)
A)
20
B)
–501
C)
21
D)
–22
Find the products AB and BA to determine whether B is the multiplicative inverse of A.
53)
A =–5–1
6 0 ,B =
01
6
–15
6
53)
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.
54)
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 .
54)
A)
ALAS
B)
ARMS
C)
ABLE
D)
ACTS
18
Perform the matrix row operation (or operations) and write the new matrix.
55)
15 –35 –10 –50
113 –3 0
2–7 4 21
1
5R1
55)
A)
3–7–2–10
113 –3 0
2–7 4 21
B)
3–7–2–50
113 –3 0
2–7 4 21
C)
3–7–2–10
1
5
13
5–3
50
2
5–7
5
4
5
21
5
D)
15 –35 –10 –50
1
5
13
5–3
50
2–7 4 21
Solve the system using the inverse that is given for the coefficient matrix.
56)
x+2y +3z =6
x+y+z= – 5
–x+y+2z = – 9
The inverse of
1 2 3
1 1 1
–1 1 2
is
1 –1–1
–3 5 2
2 –3–1
.
56)
A)
{(20, –61, 36)}
B)
{(6, –20, 18)}
C)
{(–8, –25, –12)}
D)
{(3, –4, –7)}
19
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.
57)
Adjust the contrast by leaving the black alone and changing the light grey to dark grey. Use matrix
addition to accomplish this.
57)
A)
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
B)
131
131
333
+
0 –1 0
0 –1 0
–1–1–1
=
1 2 1
1 2 1
3 2 2
C)
1 3 1
1 3 1
3 3 3
+
1 1 1
1 1 1
1 1 1
=
2 3 2
2 3 2
3 3 3
D)
131
131
333
+
1 –1 1
1 –1 1
–1–1–1
=
1 2 1
1 2 1
3 2 2
Find the products AB and BA to determine whether B is the multiplicative inverse of A.
58)
A =
1 0 0 –1
2 1 0 0
1 1 1 –2
0 0 0 1
,B =
1 0 0 1
–2 1 0–2
1 –1 1 3
0 0 0 1
58)
A)
B A–1
B)
B =A–1
20