Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
Determine which of the following matrix equations represents the solution to the system:
2x1+x2= 2
5x1+3x2= 13 .
1)
A)
x1
x2
=2
16
3–1
–5 2
B)
x1
x2
=2 1
5 3
2
13
C)
x1
x2
=2
13
–2–1
–5–3
D)
x1
x2
=3–1
–5 2
2
13
D)
Answer the question.
2)
Which of the following matrices has an inverse?
2)
A)
–2 3
4 1
B)
0 4
0–2
C)
0–1
3 5
–1 3
D)
3–2 1
4 0 7
D)
Provide an appropriate response.
3)
Given A =–1 2 –3
–4 5 6 , B =7
–2, C =9–1 5 , and D =1 0
–2 6 , determine which of the following
products is NOT defined.
3)
A)
DA
B)
AD
C)
DB
D)
BC
D)
4)
Only one of the following augmented matrices of a linear system is in a reduced form. Choose the
matrix that is in reduced form.
4)
A)
1 0 –2 2
0 0 1 3
B)
0 1 –2
1 0 –3
C)
1 4 0 4
0 0 1 –4
0 0 0 0
D)
1 0 –2
0 0 0
0 1 –1
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
5)
A chain of amusement parks pays experienced workers $240 per week and inexperienced
workers $220 per week. The total number of workers and total weekly wages at three
different parks are given in the table. How many experienced workers does each park
employ? Set up a system of linear equations and solve using matrix inverse methods.
Park 1 Park 2 Park 3
Number of workers 120 120 120
Total weekly wages 28,400 27,200 28,000
5)
Perform the operation, if possible.
6)
Let A =0–2 1
4–1 0 and B =
1–2
0 1
–2–1
. Find AB.
6)
Solve the problem.
7)
A hospital dietitian wants to insure that a certain meal consisting of rice, broccoli, and fish
contains exactly 26,800 units of vitamin A, 840 units of vitamin E, and 11,160 units of
vitamin C. One ounce of rice contains 400 units of vitamin A, 20 units of vitamin E, and 180
units of vitamin C. One ounce of broccoli contains 800 units of vitamin A, 60 units of
vitamin E, and 540 units of vitamin C. And one ounce of fish contains 2,400 units of
vitamin A, 40 units of vitamin E, and 810 units of vitamin C. How many ounces of each
food should this meal include? Set up a system of linear equations and solve using
Gauss–Jordan elimination.
7)
Provide an appropriate response.
8)
Solve the matrix equation 2 1
5 3
x1
x2
+1
5
=3
18 by using the inverse of the coefficient
matrix.
8)
9)
Solve the linear system corresponding to the following augmented matrix:
140 5
001 6
000 0
9)
10)
Use
–3 1 –1
–3 1 0
1 0 1
–1
=
1–1 1
3–2 3
–1 1 0
to solve
–3x1+x2–x3= 2
–3x1+x2= 0
x1+x3= – 3
10)
Find the system of equations to model the problem. DO NOT SOLVE THIS SYSTEM.
11)
A paper company produces high, medium, and low grade paper. The number of tons of
each grade that is produced from one ton of pulp depends on the source of that pulp. The
following table lists three sources and the amount of each grade of paper that can be made
for one ton of pulp from each source.
(Number of Tons)
High Grade Medium Grade Low grade
Brazilian Pulp 0.6 0.3 0.1
Domestic Pulp 0.5 0.3 0.2
Recycled Pulp 0.3 0.4 0.3
The paper company has orders for 11 tons of high grade, 15 tons of medium grade, and 14
tons of low grade paper. How many tons of each type of pulp should be used to fill these
orders exactly? Set up a system of linear equations, letting x, y, and z be the number of tons
of Brazilian pulp, domestic pulp, and recycled pulp, respectively, needed to fill the orders.
11)
Solve the problem.
12)
A trucking firm wants to purchase 10 trucks that will provide exactly 28 tons of additional
shipping capacity. A model A truck holds 2 tons, a model B truck holds 3 tons, and a model
C truck holds 5 tons. How many trucks of each model should the company purchase to
provide the additional shipping capacity? Set up a system of linear equations and solve
using Gauss–Jordan elimination. There may be more than one solution.
12)
13)
An economy is based on two sectors, agriculture and manufacturing. Production of a
dollar’s worth of agriculture requires an input of $0.40 from agriculture and $0.10 from
manufacturing. Production of a dollar’s worth of manufacturing requires an input of $0.20
from agriculture and $0.30 from manufacturing. Find the output for each sector that is
needed to satisfy a final demand of $16 billion for agriculture and $32 billion for
manufacturing.
13)
Provide an appropriate response.
14)
Use the matrix method on a graphing calculator to solve the system
4x1+6x2–x3+4x4= 81
x1–3x2+2x3+10x4= 95
–3x1+7x2–4x3–x4= 12
x1+x2+x3+x4= 0
Carry values to two decimal places.
14)
Perform the operation, if possible.
15)
If a and b are nonzero real numbers and A =b2 ab
–ab –a2, find A2.
15)
Solve the problem.
16)
Suppose that the supply and demand equations for a logo sweat shirt in a particular week
are p = 55 – 0.10q, for the demand equation; and p = 0.20q + 25, for the supply equation.
Find the equilibrium price and quantity.
16)
Provide an appropriate response.
17)
Use Gauss–Jordan elimination to find the inverse of 1 1
3 4 .
17)
18)
Use –7 6
1–1
–1=–1–6
–1–7 to solve
–7x1+6x2= – 5
x1–x2= – 5.
18)
Solve the system of equations by elimination.
19)
8x – 4y = 10
12x – 6y = – 15
19)
Solve the system of equations by graphing.
20)
Use a graphing utility to solve the system y = 2x + 7
y = – 5x + 1 . Give the answer to three decimal
places.
20)
Solve the problem.
21)
A supermarket chain sells oranges, apples, peaches, and bananas in three stores located
throughout a large metropolitan area. The average number of pounds sold per day in each
store is summarized in matrix M. “In season” and “out of season” prices, per pound, of each
fruit are given in matrix N. What is the total, for the three stores, of “in season” daily
revenue for the four fruits? The “out of season” peach sales represent what percentage of
the daily total “out of season” revenues for store 3?
Fruit
O A P B
M =
60 80 60 55
95 80 65 75
85 85 70 95
Store 1
Store 2
Store 3
“In season” “Out of season”
N =
$3.00 $7.00
$5.00 $9.50
$5.00 $5.50
$0.40 $0.60
O
A
P
B
21)
22)
A retail company offers, through two different stores in a city, three models, A, B, and C, of
a particular brand of camping stove. The inventory of each model on hand in each store is
summarized in matrix M. Wholesale (W) and retail (R) prices of each model are
summarized in matrix M. Find the product MN and label its columns and rows
appropriately. What is the wholesale value of the inventory in Store 1?
A B C W R
M =2 0 1
3 3 0
Store 1
Store 2 N =
$60 $90
$120 $150
$40 $50
A
B
C
22)
Find the system of equations to model the problem. DO NOT SOLVE THIS SYSTEM.
23)
If $9,000 is to be invested, part at 13% and the rest at 8% simple interest, how much should
be invested at each rate so that the total annual return will be the same as $9,000 invested
at 9%? Set up a system of linear equations, letting x1 be the amount invested at 13% and x2
be the amount invested at 8%.
23)
Use the given encoding matrix A to solve the problem.
24)
The following message was encoded with matrix 1 1
2 3 . Decode this message.
28 64 32 91 30 65 24 60 38 99 42 99 35 82 36 81 46 119 13 31 23 51
24)
Find the system of equations to model the problem. DO NOT SOLVE THIS SYSTEM.
25)
In producing three types of bricks: face bricks, common bricks, and refractory bricks, a
factory incurs labor, material, and utility costs. To produce one pallet of face bricks, the
labor, material, and utility costs are $50, $75, and $35, respectively. To produce one pallet
of common bricks, the labor, material, and utility costs are $50, $60, and $30, respectively,
while the corresponding costs for refractory bricks are $75, $100, and $45. In a certain
month the company has allocated $12,000 for labor costs, $14,500 for material costs and
$6,000 for utility costs. How many pallets of each type of brick should be produced in that
month to exactly utilize these allocations? Set up a system of linear equations, letting x, y,
and z be the number of pallets of face, common, and refractory bricks, respectively, that
must be produced in that month.
25)
Solve the problem.
26)
A company that manufactures laser printers for computers has monthly fixed costs of
$177,000 and variable costs of $650 per unit produced. The company sells the printers for
$1,250 per unit. How many printers must be sold each month for the company to break
even?
26)
Provide an appropriate response.
27)
Given matrix A:
A =
5 –3
1 7
0 –2
–91
2
What is the size of A? Find a32 and a11.
27)
28)
Given matrices M =
–1 2 0
2 3 1
–4 0 –2
, X =
x
y
z
, A =
2
4
1
, M–1=
–12
3
1
3
01
3
1
6
2–4
3–7
6
, and B =
3
0
6
solve the
matrix equations MX = A and MX = B.
28)
29)
Use a graphing utility and augmented matrix methods to solve the system:
1.8x1– 12.17x2= 33
–3.75x1+ 5.73x2= 7
Express your answer accurate to three decimal places.
29)
Solve the problem.
30)
A large oil company produces three grades of gasoline: regular, unleaded, and
super–unleaded. To produce these gasolines, equipment is used which requires as input
certain amounts of each of the three grades of gasoline. To produce a dollar’s worth of
regular requires inputs of $0.14 worth of regular, $0.18 worth of unleaded, and $0.17 worth
of super–unleaded. To produce a dollar’s worth of unleaded requires inputs of $0.14 worth
of regular, $0.15 worth of unleaded, and $0.13 worth of super–unleaded. To produce a
dollar’s worth of super–unleaded requires inputs of $0.15 worth of regular, $0.17 worth of
unleaded, and $0.11 worth of super–unleaded. In addition, the oil company has final
demands for each of the different grades of gasoline. Find the technology matrix that
would be used in determining the total output of each grade of gasoline.
30)
Provide an appropriate response.
31)
Solve the linear system corresponding to the following augmented matrix:
1 0 2 –5
0 1 –4 2
0 0 1 1
31)
Solve the problem.
32)
Given the technology matrix M and the final demand matrix D stated below, find (I – M)–1
and find the output matrix X.
M =0.3 0.3
0.2 0.2 D =70
30
32)
Perform the operation, if possible.
33)
Let A =x 5
2 y and B =4 3
8 7 . Find BA.
33)
Provide an appropriate response.
34)
Solve the matrix equation –3 4
1–2
x
y
=25
–11
by using the inverse of the coefficient matrix. Also, solve the system if the constants 25 and
–11 are replaced by 1 and 3, respectively.
34)
Find the system of equations to model the problem. DO NOT SOLVE THIS SYSTEM.
35)
Labor and material costs for manufacturing each of three types of products M, N, and P are
given in the table:
Product
M N P
Labor $50 $40 $50
Materials $60 $40 $70
The weekly allocation for labor is $50,000 and for materials is $80,000. There are to be 3
times as many units of product M manufactured as units of product P. How many of each
type of product would be manufactured each week to use exactly each of the weekly
allocations? Set up a system of linear equations, letting x1, x2, and x3 be the number of
units of products M, N, and P, respectively, manufactured in one week.
35)
Provide an appropriate response.
36)
Use Gauss–Jordan elimination, without introducing fractions, to find the inverse of
–4–2–5
0 1 0
1 0 1
.
36)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Write a system of equations associated with the augmented matrix. Do not try to solve.
37)
3 3 5 – 2
5 0 7 4
3 6 0 2
37)
A)
3x1+ 3x2+ 5x3= – 2
5x1+ 7x3= – 4
3x1+ 6x2= – 2
B)
3x1– 3x2+ 5x3= – 2
5x1+ 7x3= 4
3x1+ 6x2= 2
C)
3x1+ 3x2+ 5x3= 2
5x1+ 7x3= 4
3x1+ 6x2= 2
D)
3x1+ 3x2+ 5x3= – 2
5x1+ 7x3= 4
3x1+ 6x2= 2
Determine whether B is the inverse of A.
38)
A =
2–1 0
–1 1 –2
1 0 –1
, B =
1–1 2
–3–2 4
–1 1 1
38)
A)
Yes
B)
No
Perform the operation, if possible.
39)
A =3 0
–2 2 , B =3–2 1
0 4 -3 . Find BA.
39)
A)
BA is not defined.
B)
9-6 3
–612 -8
C)
9–6
-6 12
3-8
D)
9 0
0 8
Solve the system of equations by substitution.
40)
x + 4y =-8
-5x + 5y =-10
40)
A)
(2, 0)
B)
(0, -2)
C)
(1, -3)
D)
no solution
Solve the system mentally, without the use of a calculator or pencil–and–paper calculation. Try to visualize the graphs of
both lines.
41)
x + 0y = 9
0x + y = 4
41)
A)
x = 9; y = 2
B)
x = 2; y =9
2
C)
x = 9; y = 4
D)
x = 4; y = 9
State whether the matrix is in reduced form or not in reduced form.
42)
1 0 –1 5
0 4 1 1
0 0 1 1
42)
A)
Reduced Form
B)
Not Reduced Form
Identify the row operation that produces the resulting matrix.
43)
1 0 2
–1 1 3 1 0 2
0 1 5
43)
A)
–R2+R1
R2
B)
–R1+R2
R2
C)
R1+R2
R2
D)
R1+R2
R1
Solve the system of equations by substitution.
44)
x –3y =-1
y =2
44)
A)
(2, 5)
B)
(-7, 2)
C)
(-5, 2)
D)
(5, 2)
Use the given encoding matrix A to solve the problem.
45)
Use the given message to construct the code matrix by assigning numbers to the letters and
symbols. Use the numerical assignment a = 1, b = 2, . . . , z = 26, space = 30, period = 40, and
apostrophe = 60.
Message: CALL ME TOMORROW.
Encoding matrix A =1 0
1 1
45)
A)
B)
C)
3 1
12 12
30 14
530
18 15
13 15
19 19
15 23
40 30
D)
3 1 12
12 30 13
530 20
15 13 15
18 18 15
23 40 30
The matrix is the final matrix form for a system of two linear equations in variables x1 and x2. Write the Solution of the
system.
46)
105
013
46)
A)
x1= 5
x2= 3
B)
x1= – 5
x2= – 3
C)
x1= 5
x2= t for any real number t
D)
x1= 3
x2= 5
Solve the system mentally, without the use of a calculator or pencil–and–paper calculation. Try to visualize the graphs of
both lines.
47)
x – 0y = 5
3x + y = 7
47)
A)
x = 5; y = 4
B)
x = – 5; y = 8
C)
x = 5; y = – 8
D)
x = 3, y = 7
Perform the indicated operations given the matrices.
48)
Let A =-3 2 and B =1 0 ; 2A + 3B
48)
A)
-3 4
B)
02
C)
-6 4
D)
-5 4
Write a system of equations associated with the augmented matrix. Do not try to solve.
49)
3 0
1 1
6–5
x
y
=
–8
-8
3
49)
A)
3x + y + z = – 8
x – 5y =-8
B)
3x = – 8
x + y =-8
6x – 5y =3
C)
3x + y = – 8
x – 8y =-8
6x – 5y =3
Perform the operation, if possible.
50)
Let B =–1 3 7 –3. Find -2B.
50)
A)
2 3 7 –3
B)
2-6 -14 6
C)
-2 614 -6
D)
–3 1 5 –5
State whether the matrix is in reduced form or not in reduced form.
51)
1–5 0 0 1
0 0 120
51)
A)
Not Reduced Form
B)
Reduced Form
Perform the operation, if possible.
52)
Let A =–1 5 1 and B =
–6–2 9
–5–7–3
6 –8 2
. Find AB.
52)
A)
–13 –41 –22
B)
13 41 22
C)
6 –10 9
5 –35 –3
–6–40 2
D)
–13
–41
–22
Solve the problem.
53)
Sam and Chad are ticket–sellers at their class play. Sam is selling student tickets for $2.00 each, and
Chad selling adult tickets for $5.50 each. If their total income for 24 tickets was $83.00, how many
tickets did Sam sell?
53)
A)
14 tickets
B)
16 tickets
C)
10 tickets
D)
15 tickets
Perform the indicated row operations on the following matrix.
1–5 4
2 2 5
54)
–3R1
R1
54)
A)
1 –5 4
–6–6–15
B)
–3–5 4
–6 2 5
C)
1 –5 4
–117 7
D)
–315 –12
2 2 5
The system cannot be solved by matrix inverse methods. Find a method that could be used and then solve the system.
55)
–2x1+ 6x2= 4
6x1– 18x2= 12
55)
A)
x1= 3t + 2, x2= t for any real number t
B)
x1= 2t + 6, x2= t for any real number t
C)
x1= 3t + 2 for any real number t, x2= 0
D)
No Solution
14
Solve the equation for the indicated variable. Assume that the dimensions are such that matrix multiplication and
addition are possible and that inverses exist when needed.
56)
Solve for A: AY – A = B
56)
A)
A = Y–1 B – I
B)
A = BY–1 + I
C)
A = (Y – I)–1 B
D)
A = B(Y – I)–1
Find the inverse, if it exists, of the given matrix.
57)
A =
0 4 4
-4 0 5
0 6 0
57)
A)
Does not exist
B)
5
16 – 1
4– 5
24
0 0 1
6
1
40–
1
6
C)
– 5
16 – 1
4– 5
24
–
1
601
6
1
40 0
D)
5
16 0 1
4
– 1
40 0
– 5
24
1
6–
1
6
Write the linear system corresponding to the reduced augmented matrix.
58)
104
010
000
58)
A)
x1= 4, x2= 0
B)
x1= – 4, x2= 0
C)
No Solution
D)
x1= 4, x2= t for any real number t
Solve the system as matrix equations using inverses.
59)
x1+x2+x3= – 2
x1–x2+ 3x3= 8
5x1+x2+x3= – 22
59)
A)
(4, –5, –1)
B)
(4, –1, –5)
C)
(5, 1, 4)
D)
(–5, –1, 4)
Solve the equation for the indicated variable. Assume that the dimensions are such that matrix multiplication and
addition are possible and that inverses exist when needed.
60)
Solve for Y: XY + ZY = A
60)
A)
Y = X–1(A – ZY)
B)
Y = X–1(A – Z)
C)
Y = (X + Z)–1 A
D)
Y =A(X + Z)–1
Write the matrix equation as a system of linear equations without matrices.
61)
–5 0
1 1
-4 –5
x1
x2
=
-8
–2
–5
61)
A)
–5x1+x2+x3=-8
x1– 5x2= – 2
B)
–5x1+x2=-8
x1– 2x2= – 2
-4x1– 5x2= – 5
C)
–5x1=-8
x1+x2= – 2
-4x1– 5x2= – 5
Solve the problem.
62)
The input–output matrix for an economy is
Output:
Agri. Mfg.
Input: Agri.
Mfg.
0.04 0.18
0.02 0.22
= T
The demand matrix is D = 700
1000
Find the production matrix X.
62)
A)
X = 861.5
1108.3
B)
X =1074.2
1397.3
C)
X = 974.2
1307.0
D)
X =1207.2
985.0
Find the system of equations to model the problem. DO NOT SOLVE THIS SYSTEM.
63)
A $124,000 trust is to be invested in bonds paying 9%, CDs paying 8%, and mortgages paying 10%.
The sum of the amount invested in bonds and the amount invested in CDs must equal the
mortgage investment. To earn an $11,400 annual income from the investments, how much should
the bank invest in each?
Let x represent the amount invested in bonds, y the amount invested in CDs, and z the amount
invested in mortgages.
63)
A)
x + y – z = 11,400
x – y + 9z = 22
8x + y + z = 124,000
B)
x + y – z = 0
x + y + z = 124,000
9x + 8y + z = 11,400
C)
x + y + z = 0
x + y – 9z = 124,000
0.1x + 0.08y – 0.09z = 11,400
D)
x + y – z = 0
x + y + z = 124,000
0.09x + 0.08y + 0.1z = 11,400
Solve the system of equations by elimination.
64)
x – 5y = 35
5x – 4y = 28
64)
A)
(1 , – 7)
B)
(1, – 8)
C)
(7, 0 )
D)
(0, – 7)
Solve the problem.
65)
A Dawn Bakery bakes whole wheat, oat, and rye bread, with mixing, baking, and packaging times,
in hours, as shown:
Mix Bake Package
A =
0.04 0.07 0.02
0.03 0.05 0.02
0.04 0.06 0.02
Whole wheat
Oat
Rye
An order is received for 400 loaves of whole wheat bread, 200 loaves of oat bread, and 350 loaves of
rye bread. Given that the cost of mixing, baking, and packaging is $14, $25, and$2, respectively, per
hour, find matrices B and C so that the product BAC will give the total cost (excluding raw
materials) of filling this order. Find the total cost.
65)
A)
B =
14
25
2
, C =400 200 350 , total cost = $1313
B)
B =400 200 350 , C =14 25 2 , total cost = $2017
C)
B =400 200 350 , C =
14
25
2
, total cost = $2017
D)
B =14 25 2 , C =
400
200
350
, total cost = $1313
Find the matrix product mentally, without the use of a calculator or pencil–and–paper calculations.
66)
100
010
001
1 2 3
4 5 6
7 8 9
66)
A)
1 2 3
4 5 6
7 8 9
B)
9 8 7
6 5 4
3 2 1
C)
1
D)
1 –2 3
–4 5 –6
7 –8 9
Solve the system of equations by elimination.
67)
x + y =8
x – y =8
67)
A)
(0, 8)
B)
(8, 0)
C)
(-8, 0)
D)
(0, -8)
The system cannot be solved by matrix inverse methods. Find a method that could be used and then solve the system.
68)
–2x1+ 6x2= – 4
6x1– 18x2= 12
68)
A)
x1= 3t + 2 for any real number t, x2= 0
B)
x1= 2t + 6, x2= t for any real number t
C)
No Solution
D)
x1= 3t + 2, x2= t for any real number t
Write the matrix equation as a system of linear equations without matrices.
69)
3 3 5
5 0 7
3 6 0
x1
x2
x3
=
–2
4
2
69)
A)
3x1+ 3x2+ 5x3= 2
5x1+ 7x3= 4
3x1+ 6x2= 2
B)
3x1+ 3x2+ 5x3= – 2
5x1+ 7x3= – 4
3x1+ 6x2= – 2
C)
3x1+ 3x2+ 5x3= – 2
5x1+ 7x3= 4
3x1+ 6x2= 2
D)
3x1– 3x2+ 5x3= – 2
5x1+ 7x3= 4
3x1+ 6x2= 2
Perform the operation, if possible.
70)
5 4 +– 2
7
70)
A)
3 11
B)
3
11
C)
5– 2
4 7
D)
Not defined
Provide an appropriate response.
71)
Write the augmented matrix for the system.
6x1+ 4x2= 30
8x2= 72
71)
A)
30 4 6
72 0 8
B)
6 4 30
0 8 72
C)
8 0 72
6 4 4
D)
6 4 30
8 72 0
State whether the matrix is in reduced form or not in reduced form.
72)
1 3 0 1
0 0 1 0
0 0 0 1
72)
A)
Not Reduced Form
B)
Reduced Form
Perform the indicated row operations on the following matrix.
1–5 4
2 2 5
73)
(–2)R1+R2
R2
73)
A)
1–5 4
012 –3
B)
–210 –8
2 12 –5
C)
–210 –8
0 12 –3
D)
012 –3
2 2 5
Identify the row operation that produces the resulting matrix.
74)
2 0 6
–2 2 8 2 0 6
0 1 7
74)
A)
R1+R2
R2
B)
1
2R2
R1
C)
1
2R1+1
2R2
R2
D)
–1
2R1+ – 1
2R2
R1