67)
Let C =
1
– 3
2
and D =
– 1
3
– 2
. Find C – 4D.
A)
–3
9
–6
B)
– 5
15
–10
C)
5
–6
4
D)
5
–15
10
Answer:
D
68)
A =3–2
4 0 , B =0–2
4 6 Find AB.
A)
0 4
16 0
B)
–8–18
0–8
C)
–8 0
36 –8
D)
–18 –8
–8 0
Answer:
B
69)
Let A =–1 5 1 and B =
–6–2 9
–5–7–3
6 –8 2
. Find AB.
A)
–13
–41
–22
B)
–13 –41 –22
C)
13 41 22
D)
6 –10 9
5 –35 –3
–6–40 2
Answer:
B
70)
A =3 0
–2 2 , B =3–2 1
0 4 –3. Find BA.
A)
9–6 3
–612 –8
B)
BA is not defined.
C)
9–6
–612
3–8
D)
9 0
0 8
Answer:
B
71)
Let A =0–2 1
4–1 0 and B =
1–2
0 1
–2–1
. Find AB.
Answer:
–2–3
4–9
72)
Let A =x 5
2 y and B =4 3
8 7 . Find BA.
Answer:
4x + 6 20 + 3y
8x + 14 40 + 7y
73)
If a and b are nonzero real numbers and A =b2 ab
–ab –a2, find A2.
Answer:
b4–a2b2ab3–a3b
–ab3+a3b–a2b2+a4
74)
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.
A)
DB
B)
BC
C)
DA
D)
AD
Answer:
D
75)
Let A =3 3
2 4 and B =0 4
–1 6 ; 2A + B
A)
610
110
B)
614
220
C)
6 7
310
D)
610
314
Answer:
D
76)
Let C =
1
–3
2
and D =
–1
3
–2
; C – 3D
A)
–2
6
–4
B)
4
–12
8
C)
–4
12
–8
D)
4
–6
4
Answer:
B
77)
Let A =–2 2 and B =1 0 ; 2A + 3B
A)
–1 4
B)
1 2
C)
–4 4
D)
–3 4
Answer:
A
78)
a b
c d
+ 0 5
–4 7
=3 7
2–1
A)
–3–2
–6 8
B)
0 2
6 8
C)
3 12
–2 6
D)
3 2
6–8
Answer:
D
79)
1 2
0–1
a b
c d
=1 2
2 4
A)
0 0
2 5
B)
2 1
4 2
C)
5 10
–2–4
D)
1 0
0 1
Answer:
C
80)
A company makes three chocolate candies: cherry, almond, and raisin. Matrix A gives the number of units of
each ingredient in each type of candy in one batch. Matrix B gives the cost of each ingredient (dollars per unit)
from suppliers X and Y. What is the cost of 100 batches from supplier X?
A)
$3300
B)
$6600
C)
$7800
D)
$4800
Answer:
C
81)
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 t
product BAC will give the total cost (excluding raw materials) of filling this order. Find the total cost.
A)
B =400 200 350 , C =14 25 2 , total cost = $2017
B)
B =14 25 2 , C =
400
200
350
, total cost = $1313
C)
B =400 200 350 , C =
14
25
2
, total cost = $2017
D)
B =
14
25
2
, C =400 200 350 , total cost = $1313
Answer:
C
82)
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
Answer:
$3,010; 20.87%
83)
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
Answer:
W R
$160 $230
$540 $720
Store 1
Store 2
$160
84)
1 0
0 1
2 3
5 1
A)
3 2
1 5
B)
1
2
1
3
1
51
C)
1 1
1 1
D)
2 3
5 1
Answer:
D
85)
1 0 0
0 1 0
0 0 1
1 2 3
4 5 6
7 8 9
A)
1
B)
9 8 7
6 5 4
3 2 1
C)
1 –2 3
–4 5 –6
7 –8 9
D)
1 2 3
4 5 6
7 8 9
Answer:
D
86)
A =5 3
3 2 , B =2–3
–3 5
A)
Yes
B)
No
Answer:
A
87)
A =
2–1 0
–1 1 –2
1 0 –1
, B =
1–1 2
–3–2 4
–1 1 1
A)
Yes
B)
No
Answer:
B
88)
Which of the following matrices has an inverse?
A)
0–1
3 5
–1 3
B)
3–2 1
4 0 7
C)
–2 3
4 1
D)
0 4
0–2
Answer:
C
89)
5 8
3 5
A)
5–8
–3 5
B)
–5–8
–3–5
C)
–5 3
8–5
D)
5 3
8 5
Answer:
A
90)
–6–6
–5–5
A)
–5
11 –6
11
–5
11 –6
11
B)
–5
11
6
11
5
11 –6
11
C)
5
11 –6
11
–5
11
6
11
D)
Does not exist
Answer:
D
91)
A =
0 4 4
–5 0 7
0 3 0
A)
7
20 0 1
4
–1
50 0
–7
15
1
3–1
3
B)
–7
20 –1
5–7
15
–1
301
3
1
40 0
C)
Does not exist
D)
7
20 –1
5–7
15
0 0 1
3
1
40–1
3
Answer:
D
92)
1 1 1
2 1 1
2 2 3
A)
–1–1–1
–2–1–1
–2–2–3
B)
1 1 1
1
2 1 1
1
2
1
2
1
3
C)
–1 1 0
4–1–1
–2 0 1
D)
Does not exist
Answer:
C
93)
Use Gauss–Jordan elimination to find the inverse of 1 1
3 4 .
Answer:
4–1
–3 1
94)
Use Gauss–Jordan elimination, without introducing fractions, to find the inverse of
–4–2–5
0 1 0
1 0 1
.
Answer:
1 2 5
0 1 0
–1–2–4
95)
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
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
Answer:
B
96)
A message has been encoded and the matrix which the receiver gets is shown below.
The encoding matrix A which was used to encode the message is:
A =0 2
3 1
Find the decoding matrix A–1, and use it to decode the message.
Assume that the numerical assignment used was a = 1, b = 2, ….., z = 26, space = 30, period = 40, and apostrophe
60.
A)
DRINK ENOUGH MILK
B)
EAT YOUR VEGETABLES
C)
DRINK ENOUGH COKE
D)
EAT YOUR BROCCOLI
Answer:
B
97)
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
Answer:
THE YELLOW OWL IS HERE
98)
6 0
1 1
–3–5
x1
x2
=
1
5
6
A)
6x1+x2=1
x1+ 5x2=5
–3x1– 5x2=6
B)
6x1=1
x1+x2=5
–3x1– 5x2=6
C)
6x1+x2+x3=1
x1– 5x2=5
Answer:
B
99)
3 3 5
5 0 7
3 6 0
x1
x2
x3
=
–2
4
2
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
Answer:
A
100)
8x1+ 9x2= 117
4x1+ 6x2= 66
A)
8 9
4 6
x1
x2
=117
66
B)
8 9
6 4
x1
x2
=66
117
C)
117 9
66 6
x1
x2
=8
6
D)
8 4
9 6
x1
x2
=117
66
Answer:
A
101)
6x1+ 4x2= 30
8x2= 72
A)
8 0
6 4
x1
x2
=72
4
B)
30 4
72 0
x1
x2
=6
8
C)
6 4
0 8
x1
x2
=30
72
D)
6 4
872
x1
x2
=30
0
Answer:
C
102)
Solve for C: CZ = Y
A)
C = YZ–1
B)
C = ZY–1
C)
C =Z–1Y
D)
C = Y/Z
Answer:
A
103)
Solve for A: AY – A = B
A)
A = (Y – I)–1 B
B)
A = B(Y – I)–1
C)
A = BY–1 + I
D)
A = Y–1 B – I
Answer:
B
104)
Solve for Y: XY + ZY = A
A)
Y = X–1(A – Z)
B)
Y = (X + Z)–1 A
C)
Y = X–1(A – ZY)
D)
Y =A(X + Z)–1
Answer:
B
105)
–5x1+ 3x2= 8
3x1– 6x2= – 30
A)
(–6, –2)
B)
(–2, –6)
C)
(6, 2)
D)
(2, 6)
Answer:
D
106)
– 2x1+ 6x2= 6
3x1+ 2x2= 13
A)
(–3 , – 2)
B)
(3,2)
C)
(2,3)
D)
(–2, –3)
Answer:
B
107)
x1+x2+x3= – 2
x1–x2+ 3x3= 8
5x1+x2+x3= – 22
A)
(5, 1, 4)
B)
(4, –5, –1)
C)
(–5, –1, 4)
D)
(4, –1, –5)
Answer:
C
108)
Determine the value of each variable.
x + 3 y + 4
7 –1
=6 0
7 k
A)
x = – 3
y = 4
k = – 1
B)
x = 6
y = 0
k = – 1
C)
x = 3
y = – 4
k = – 1
D)
x = – 3
y = – 4
k = – 1
Answer:
C
109)
Determine which of the following matrix equations represents the solution to the system:
2x1+x2= 2
5x1+3x2= 13 .
A)
x1
x2
=2 1
5 3
2
13
B)
x1
x2
=3–1
–5 2
2
13
C)
x1
x2
=2
13
–2–1
–5–3
D)
x1
x2
=2
16
3–1
–5 2
Answer:
B
110)
Use –7 6
1–1
–1=–1–6
–1–7 to solve
–7x1+6x2= – 5
x1–x2= – 5.
Answer:
x1= 35; x2= 40
111)
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
Answer:
x1= – 1, x2= – 3, x3= – 2
112)
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.
Answer:
x = – 3, y = 4; x = – 7, y = – 5
113)
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.
Answer:
x = 1, y = 1.5, z = – 2.5; x =- 1, y = 1, z = – 1
114)
Solve the matrix equation 2 1
5 3
x1
x2
+1
5
=3
18 by using the inverse of the coefficient matrix.
Answer:
x1= – 7, x2= 16
115)
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.
Answer:
x1= 8.85, x2= – 2.78, x3= – 17.31, x4=11.24
116)
–2x1+ 6x2= – 4
6x1– 18x2= 12
A)
No Solution
B)
x1= 3t + 2, x2= t for any real number t
C)
x1= 3t + 2 for any real number t, x2= 0
D)
x1= 2t + 6, x2= t for any real number t
Answer:
B
117)
–2x1+ 6x2= 4
6x1– 18x2= 12
A)
x1= 3t + 2, x2= t for any real number t
B)
x1= 3t + 2 for any real number t, x2= 0
C)
x1= 2t + 6, x2= t for any real number t
D)
No Solution
Answer:
D
118)
There were 340 people at a play. The admission price was $2 for adults and $1 for children. The admission
receipts were $490. How many adults and how many children attended?
A)
150 adults and 190 children
B)
95 adults and 245 children
C)
122 adults and 218 children
D)
190 adults and 150 children
Answer:
A
119)
A company produces three models of MP3 players, models A, B, and C. Each model A machine requires
3.2 hours of electronics work, 2.8 hours of assembly time, and 4.4 hours of quality assurance time. Each model B
machine requires 5.4 hours of electronics work, 2.4 hours of assembly time, and 3.4 hours of quality assurance
time. Each model C machine requires 2.2 hours of electronics work, 5.8 hours of assembly time, and 4.8 hours of
quality assurance time. There are 303 hours available each week for electronics, 393 hours for assembly, and
416 hours for quality assurance. How many of each model should be produced each week if all available time
must be used?
A)
Model A: 31
Model B: 20
Model C: 44
B)
Model A: 28
Model B: 22
Model C: 45
C)
Model A: 30
Model B: 15
Model C: 50
D)
Model A: 30
Model B: 20
Model C: 45
Answer:
D
120)
Your screen print operation is doing extremely well at the craft shows. Last week you sold 50 tie–dyed shirts for
$15 each, 40 Cheraw–Tech crew shirts for $10 each and 30 handpainted T–shirts for $12 each. Use matrix
operations to calculate your total revenue for the week.
A)
$1480
B)
$1510
C)
$1151
D)
$1750
Answer:
B
121)
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
Answer:
Park 1: 100 experienced workers
Park 2: 40 experienced workers
Park 3: 80 experienced workers
122)
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.
A)
X = 861.5
1108.3
B)
X =1207.2
985.0
C)
X = 974.2
1307.0
D)
X =1074.2
1397.3
Answer:
C
123)
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 internal consumption.
A)
161.5
108.3
B)
374.2
397.3
C)
507.2
57.0
D)
274.2
307.0
Answer:
D
124)
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
Answer:
(I –M)–1=1.6 0.6
0.4 1.4 X =130
70
125)
A textbook economy has only two industries, the electric company and the gas company. Each dollar‘s worth of
the electric company’s output requires 0.20 of its own output and 0.4 of the gas company’s output. Each dollar’s
worth of the gas company’s output requires 0.50 of its own output and 0.7 of the electric company’s output.
What should the production of electricity and gas be (in dollars) if there is a $16 M demand for electricity and a $7
M demand for gas?
A)
Electricity: $107.5 M; Gas: $100 M
B)
Electricity: $125 M; Gas: $92.5 M
C)
Electricity: $115 M; Gas: $103.5 M
D)
Electricity: $97.5 M; Gas: $103 M
Answer:
A
126)
Two sectors of a textbook economy are (1) communication equipment and (2) components and accessories. In
2005 the input–output table involving these two sectors was as follows.
To Equipment Components
From Equipment 6,000 500
Components 24,000 30,000
Total Output 90,000 140,000
Determine the production levels necessary in these two sectors to meet a demand for $80,000 of equipment and $90,000
of components. Round to significant digits.
A)
Equipment: 86,000
Components: 90,000
B)
Equipment: 90,000
Components: 140,000
C)
Equipment: 24,000
Components: 140,000
D)
Equipment: 86,000
Components: 140,000
Answer:
D
127)
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.
Answer:
$44 billion agriculture, $52 billion manufacturing
128)
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.
Answer:
M =
0.14 0.14 0.15
0.18 0.15 0.17
0.17 0.13 0.11