Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
If the orders of the matrices Q, R, and S are 4 × 5, 3 × 2, and 5 × 3, respectively, then the order of
QRS is
1)
A)
4 × 5.
B)
3 × 5.
C)
2 × 4.
D)
3 × 3.
E)
none of the above
2)
Consider the matrix
5–9–1
0 7 –1
0 0 3
, this matrix can be best described as a(n):
2)
A)
lower triangular matrix
B)
upper triangular matrix
C)
diagonal matrix
D)
main diagonal
3)
If x1–x2–3x3–4x4=3
3x1+x2–x3+4x4=5, then
3)
A)
x1= 2 +x3, x2= – 1 – 2x3– 4x4, x3=x3, x4=x4
B)
x1= 3 – 6x4, x2= 4 –x4, x3= – 2 + 3x4, x4=x4
C)
x1= 3 – 6x4, x2= 4 +x4, x3= – 2 + 3x4, x4=x4
D)
x1= – 1 +x3–x4, x2= 4 + 2x3+x4, x3=x3, x4=x4
E)
x1= – 1 +x4, x2= 3x4, x3= 6 + 2x4, x4=x4
4)
If R=
0–1 0
1 2 –1
1 1 0
, then R–1=
4)
A)
2–1 2
–1 1 3
4 2 –2
B)
3
2
5
2–3
2
–14
3
1
2
22
3–5
3
C)
1 0 1
–1 0 0
–1–1 1
D)
1 0 –3
–1 2 4
0 1 6
E)
none of the above
5)
If A=aij where A is 2 × 2 and aij =i– 2j, then matrix A is given by
5)
A)
–1–4
–1–3.
B)
–1–3
0–2.
C)
2–1
3 0 .
D)
–1 0
–3–2.
E)
2 1
3 4 .
6)
If
x–2y–4z=4
2x+y+z=9
x+y–z=1
, then
6)
A)
x= 2z, y= 3 – 4z, and z=t
B)
x= 5, y= – 5
2, and z=3
2
C)
x= 4 –13
2z, y=11
2–1
6z, and z= t
D)
x=22
5, y=1
5, and z= 0
E)
none of the above
7)
If
x+y–3z=5
x–3y+z= – 7
2x–y–3z=2
, then
7)
A)
x= 2 – 4z, y= – 5 + 2z, and z= t.
B)
x= 3, y= – 1, and z= 4.
C)
x= 5 – 3z, y= 3 + 4z, and z=t.
D)
x= 5, y= – 8, and z= 12.
E)
none of the above
8)
If 3x–y
x y
=3 2
2x y, then y=
8)
A)
–4.
B)
–2.
C)
4.
D)
0.
E)
2.
9)
If 4 1 x
–2 0
+ 2 –2 0
y0
=0 0
0 0 , then
9)
A)
x= 4 and y= – 5.
B)
x= 0 and y= 0.
C)
x= 0 and y= 4.
D)
x=1
4and y=1
2.
E)
There are no values for x and y which satisfy the equation.
10)
Reducing
2 2 4
1 1 2
1 0 1
gives
10)
A)
1 0 1
0 1 2
0 0 0
B)
1 1 2
0 0 0
1 0 1
C)
1 0 3
0 0 1
0 0 0
D)
1 0 0
0 1 0
0 0 1
E)
1 0 1
0 1 1
0 0 0
11)
1–1
3 4
0 2
5–3
=
11)
A)
0–2
15 –12
B)
6 8
–4–17
C)
–5 5
20 –6
D)
–2 8
3 8
E)
4–1
0 6
12)
1–1 2
4
–1
0
2 3 +
1 0
0 1
–1–1
=
12)
A)
2 9 –3
B)
2–3
4 8
1 0
C)
5
–7
D)
912
E)
none of the above
13)
21 0 –4
2–3–1
–4–2 0
3–5 1
=
13)
A)
–3 2 –4
–1 2 –2
B)
–2–2–8
1–11 –3
C)
–2 2 –8
1 –1–3
D)
–6 4 –8
–2 4 –4
E)
none of the above
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
14)
An appliance store has 25 refrigerators, 30 ranges, and 10 dishwashers in stock. If the value
of each refrigerator is $600, each range is $300 and each dishwasher is $250, use matrix
operations to find the total value of the appliance store’s inventory.
14)
5
15)
Let A=1 2
3 4 ; find A–1.
15)
16)
If A=
–8 3
2 1
1–7
and B=
5 2
–2 9
4–3
, find A–B.
16)
17)
Solve by the method of reduction:
2x – 5y= 10
3x–y= 2
17)
18)
If A=
5–2
–4 3
1 4
, B=
7 5 0
–2 1 1
1–3–5
, and C=
2–4
5 7
–4 9
, find 3C + BA.
18)
19)
Find the inverse matrix of 2 1
–2 3 .
19)
20)
Perform the indicated operations and simplify your answer: 2 –31
6
20)
6
21)
A pharmaceutical company manufactures 3 drugs for high blood pressure, A, B, and C,
each of which requires a 2–step production process. The production cost of each drug has a
materials and labor component. If the matrices S1 and S2 represent the costs per ounce
associated with each step, use the matrix operations on a graphing calculator to write a
matrix that shows the total cost of the two steps.
Materials
Labor S1=2.00 1.50 1.80
1.25 1.75 1.50 ; S2=1.50 1.25 2.00
2.50 1.75 1.00
21)
22)
Without solving, determine whether the following system of linear equations has a unique
solution, or infinitely many solutions.
3x+ 5y– 11z= 0
9x+ 13y– 15z= 0
22)
23)
Reduce the matrix: 1 2 0
3–5 2
23)
Secret messages can be encoded by using a code and an encoding matrix. If we have the code:
a b c d e f g h i j
1 2 3 4 5 6 7 8 9 10
k l m n o p q r s t
11 12 13 14 15 16 17 18 19 20
u v w x y z space
21 22 23 24 25 26 27
and an encoding matrix: E =1 3
2 4 , we can encode a message by taking every two letters of the message, converting them
to their corresponding numbers, creating a 2 × 1 matrix, and then multiplying each two numbers by E. Use the code and
encoding scheme above and an additional encoding matrix, F =5 7
6 8 , to answer the question.
24)
If the letters to be encoded are L=n
o, show that E(FL) = (EF)L.
24)
Provide an appropriate response.
25)
Write a diagonal matrix of size 3 × 3 with entries aii =i+ 3, aij = 0 for ij.
25)
7
26)
If A=
1 0 3
–1 2 1
0 1 –1
, use row reduction to determine A–1 providing it exists.
26)
27)
A small airline has 3 flights, A, B, and C, to each of 3 cities in Colorado. The matrix N
represents the number of passengers carried in November, and matrix D represents the
number of passengers carried in December. Write a matrix that shows the total number of
passengers carried in these two months.
Denver
Boulder
Aspen
N=
2000 3000 1500
4000 1000 3500
1500 2000 2500
; D=
2500 3500 2000
4500 1500 4500
2000 3000 3000
27)
An air freight company has three types of aircraft which carry three types of cargo. The payload, in tons, is summarized
in the table below.
Passenger Transport Commuter
Mail
Medical
Freight
2
1
3
4
2
2
1
1
0
28)
Each day of the holiday season, the company must move 75 tons of mail, 50 tons of medical
supplies, and 55 tons of freight. This system of equations can be represented by the matrix:
2 4 1 75
1 2 1 50
3 2 0 55
. Reduce this matrix to calculate how many aircraft of each type should be
used.
28)
8
Provide an appropriate response.
29)
If A is a 3 × 5 matrix, B is a 5 × 7 matrix, C is a 7 × 11 matrix, D is a 5 × 7 matrix, which of
the following matrix products are defined? Give the size of each answer if it is defined.
(a) A×B×C
(b) B×C×D
(c) A×D×C
(d) A×B×D
29)
A women’s clothing chain takes inventory of one brand of sweater. The sweaters come in 3 sizes: small, medium, and
large, and 5 colors. The inventories at stores A, B, and C are represented by the matrices below.
Green
Blue
Black
Pink
Gold
A =
4 2 8
6 0 7
9 1 3
0 3 2
5 0 7
; B =
3 8 2
7 0 1
3 4 2
6 8 5
0 2 4
; C =
5 1 0
3 5 9
7 4 1
0 7 6
3 2 0
30)
Use the matrix operations on a graphing calculator to show that A+ (B+C) = (A+B) +C.
30)
Provide an appropriate response.
31)
Solve by the method of reduction:
5x + 3y + z= 2
2x–y+ 2z= 10
4x– 2y+ 3z= 17
31)
32)
A manufacturer who produces one product is only interested in tracking material costs.
What is the order of the matrix he would use?
32)
33)
Solve by the method of reduction:
x + 2y– 5z= 1
3x – 4y– 11z= 6
–2x– 6y– 16z= – 5
33)
34)
Write A=aij if A is 2 × 3 and aij = 2i+j.
34)
Suppose that an automobile manufacturer has accepted orders for 30 minivans, 25 sport utility vehicles, and 15 sedans.
These orders can be represented by the row vector Q = [ 30 25 15 ].
The “raw materials” that go into each type of vehicle are steel, glass, plastic, paint, and labor. The entries in matrix R
below give the number of units of each raw material (in this order) which are needed for each type of vehicle.
R =
6 3 9 5 4
9 4 7 4 6
5 2 6 3 3
Minivan
SUV
Sedan
.
Suppose that steel costs $800 per unit, glass costs $400 per unit, plastic costs $300 per unit, paint costs $200 per unit, and
labor costs $1000 per unit. This data can be written as the column cost vector C =
800
400
300
200
1000
.
The price the manufacturer negotiated for each minivan is $16,000, for each SUV is $21,000, and for each sedan is
$13,000. This information can be written as the column price vector P =
16,000
21,000
13,000
.
35)
Find the total cost of raw materials for these vehicles.
35)
Provide an appropriate response.
36)
Perform the indicated operation if possible:
4
8
0
2 6 5
4–1 3
36)
37)
An electronics store has 35 televisions, 15 VCRs, and 25 CD players in stock, and a second
store with 45 televisions, 25 VCRs, and 30 CD players in stock. If the value of each
television is $400, each VCR is $200 and each CD player is $150, find the total value of the
electronics store’s inventory.
37)
38)
The price charged for 2 different CDs at two different stores can be represented by the
matrix P=14 16
13 15
Store A
Store B . The quantities of each CD sold at each store can be
represented by the matrix Q=20 30
25 40 . Show that the transpose of the income generated
(PQ)T is equal to the product of the transposes of P and Q in reverse order, QTPT.
38)
Let matrix A represent the sales (in thousands of dollars) of a toy company in 1994 in three cities, let B represent the sales
in the same cities in 1995, and let C represent the sales in the same cities in 1996.
Action
Educational A =400 350 150
450 280 850 ; B =410 300 200
375 300 710 ; C =380 330 220
460 320 750
39)
What is the change in sales between 1995 and 1996?
39)
Provide an appropriate response.
40)
Solve the matrix equation: 3
x
y
z
– 2
1
–2
3
=
1
10
3
40)
A plane in three dimensional space can be written as ax +by +cz =d. We can find the possible intersections of planes in
this form by writing them as systems of linear equations and using reduction to solve them. If d= 0 in each equation,
then we have a homogeneous system with either a unique solution or infinitely many solutions.
41)
Determine whether the intersection of the planes:
2x+y+z=0
5x+4y+5z=0
x+2y+3z=0
has a unique solution or infinitely many solutions; then solve the system.
41)
An air freight company has three types of aircraft which carry three types of cargo. The payload, in tons, is summarized
in the table below.
Passenger Transport Commuter
Mail
Medical
Freight
2
1
3
4
2
2
1
1
0
42)
On Sundays and holidays, , the company must move 14 tons of mail, 9 tons of medical
supplies, and 11 tons of freight. This system of equations can be represented by the matrix:
2 4 1 14
1 2 1 9
3 2 0 11
. Reduce this matrix to calculate how many aircraft of each type should be
used.
42)
Provide an appropriate response.
43)
Solve the following system of equations by reducing the matrix:
2x+y=11
9x+5y=7
43)
44)
Find the transpose of the matrix:
1 3 3
4 5 6
7 8 9
44)
12
45)
If A=
–8 3
2 1
1–7
and B=
5 2
–2 9
4–3
, find –4A + 5B.
45)
Use the encoding matrix E =1 3
2 4 , an additional coding matrix, F =
–23
2
1–1
2
, plus the matrices I =1 0
0 1 and O =0 0
0 0
to answer the question.
46)
EO
46)
An investment firm offers 3 stock portfolios, A, B, and C. The number of blocks of each type of stock in each of these
portfolios is summarized in the following table:
Portfolio
A B C
High 6 1 3
Risk: Moderate 3 2 3
Low 1 5 3
47)
A client wants 27 blocks of high–risk stock, 27 blocks of moderate–risk stock, and 41 blocks
of low–risk stock. Use the matrix operations on a graphing calculator to find how many of
each portfolio should be suggested.
47)
The energy economy of an area is composed of four industries: gas, coal, hydroelectric and nuclear. The three main
consumers of energy are area residential consumers, a manufacturing plant, and a university. Assume that each consumer
may use some of the energy produced by each industry, and also that each industry uses some of the energy produced by
each of the other industries. The energy needs of each consumer and industry are represented by a demand vector whose
entries, in order, give the amount of gas, coal, hydroelectric, and nuclear energy needed by each consumer or industry, in
some convenient units. The demand vectors for the consumers are: DR= [ 2 4 0 1 ]; DM= [ 3 0 2 5 ];
DU= [ 0 3 5 1 ]; and the demand vectors for the industries are: DG= [ 0 1 3 2 ]; DC= [ 4 0 1 2 ]; DH= [ 3 1 0 1 ];
DN= [ 2 1 3 0 ].
48)
What is the total demand for the four types of energy from only the energy industries?
48)
Provide an appropriate response.
49)
(a) If A is the coefficient matrix of the system
x+ 2y + 3z= 1
2x + 5y + 7z= 9
–3x– 6y– 10z= 8
, determine A–1.
(b) Use A–1 to solve the system.
49)
50)
Solve the following system of equations by reduction.
2x+ 3y+ 4z+ 9u= 4
3x+ 4y+ 8z+ 7u= 1
50)
Secret messages can be encoded by using a code and an encoding matrix. If we have the code:
a b c d e f g h i j
1 2 3 4 5 6 7 8 9 10
k l m n o p q r s t
11 12 13 14 15 16 17 18 19 20
u v w x y z space
21 22 23 24 25 26 27
and an encoding matrix: E =1 3
2 4 , we can encode a message by taking every two letters of the message, converting them
to their corresponding numbers, creating a 2 × 1 matrix, and then multiplying each two numbers by E. Use the code and
encoding scheme above and an additional encoding matrix, F =5 7
6 8 , to answer the question.
51)
If the letters to be encoded are L=i
t, show that E(FL) = (EF)L.
51)
Set up a matrix equation with integer values that is equivalent to the system of equations.
52)
y= – 3
4x+ 3
y= – 1
5x+9
5
52)
14
Use the encoding matrix E =1 3
2 4 , an additional coding matrix, F =
–23
2
1–1
2
, plus the matrices I =1 0
0 1 and O =0 0
0 0
to answer the question.
53)
EF
53)
Provide an appropriate response.
54)
If A=1 2 3
3 4 5 ; B=2–1–2
1–3–2, then find (A+B)T.
54)
55)
Suppose that a simple economy consists of three sectors: agriculture (A), manufacturing
(M), and transportation (T). Economists have determined that to produce one unit of A
requires 1
10 units of A, 2
5 units of B, and 2
5 units of T, while production of one unit of M
requires 1
4 units of A, 3
16 units of M, and 1
8 units of T, and production of one unit of T
requires 2
15 units of A, 1
3 units of M, and 1
3 units of T. There is an external demand for 20
units of A, 40 units of M, and 10 units of T. Determine the production levels necessary to
meet the external demand.
55)
Suppose that an automobile manufacturer has accepted orders for 30 minivans, 25 sport utility vehicles, and 15 sedans.
These orders can be represented by the row vector Q = [ 30 25 15 ].
The “raw materials” that go into each type of vehicle are steel, glass, plastic, paint, and labor. The entries in matrix R
below give the number of units of each raw material (in this order) which are needed for each type of vehicle.
R =
6 3 9 5 4
9 4 7 4 6
5 2 6 3 3
Minivan
SUV
Sedan
.
Suppose that steel costs $800 per unit, glass costs $400 per unit, plastic costs $300 per unit, paint costs $200 per unit, and
labor costs $1000 per unit. This data can be written as the column cost vector C =
800
400
300
200
1000
.
The price the manufacturer negotiated for each minivan is $16,000, for each SUV is $21,000, and for each sedan is
$13,000. This information can be written as the column price vector P =
16,000
21,000
13,000
.
56)
Use the matrix operations on a graphing calculator to find the cost of each type of vehicle.
56)
Provide an appropriate response.
57)
If A=
2 7 4
1 9 5
3 6 8
, determine (a) a23, and (b) the order of A.
57)
58)
Solve the matrix equation: xy – 1
5 2x
=2y4
5 2x
58)
59)
Let A=1 1
2–1; B=3–5
–9 2 ; C=–2 2
4–1. Find AB+ 2C
59)
16
The energy economy of an area is composed of four industries: gas, coal, hydroelectric and nuclear. The three main
consumers of energy are area residential consumers, a manufacturing plant, and a university. Assume that each consumer
may use some of the energy produced by each industry, and also that each industry uses some of the energy produced by
each of the other industries. The energy needs of each consumer and industry are represented by a demand vector whose
entries, in order, give the amount of gas, coal, hydroelectric, and nuclear energy needed by each consumer or industry, in
some convenient units. The demand vectors for the consumers are: DR= [ 2 4 0 1 ]; DM= [ 3 0 2 5 ];
DU= [ 0 3 5 1 ]; and the demand vectors for the industries are: DG= [ 0 1 3 2 ]; DC= [ 4 0 1 2 ]; DH= [ 3 1 0 1 ];
DN= [ 2 1 3 0 ].
60)
Use the matrix operations on a graphing calculator to find the total demand for the four
types of energy from only the consumers.
60)
Provide an appropriate response.
61)
An appliance store has 25 refrigerators, 30 ranges, and 10 dishwashers in stock, and a
second store with 15 refrigerators, 25 ranges, and 20 dishwashers in stock. If the value of
each refrigerator is $600, each range is $300 and each dishwasher is $250, find the total
value of the inventory at the two appliance stores.
61)
In the energy demand problem above, the demand vectors could be represented in a 2 row matrix, where row 1 contains
the energy demands of the consumers and row 2 contains the energy demands of the energy industries:
D =5 7 7 7
9 3 7 5
62)
The researcher then changes the original demand matrix using the matrix equation D2=A
+ 0.2D, where A=1 2 1 2
2 1 2 1 . Use the matrix operations on a graphing calculator to
calculate the new demand matrix in his model.
62)
Secret messages can be encoded by using a code and an encoding matrix. If we have the code:
a b c d e f g h i j
1 2 3 4 5 6 7 8 9 10
k l m n o p q r s t
11 12 13 14 15 16 17 18 19 20
u v w x y z space
21 22 23 24 25 26 27 and an encoding matrix E, we can encode a
message by taking every two letters of the message, converting them to their corresponding numbers, creating a 2 × 1
matrix, and then multiplying each two numbers by E. The message may be unscrambled with a decoding matrix which is
the inverse of the coding matrix, E–1. Determine if the given pair of encoding matrices are inverses of each other.
63)
6 8
5 7 and 3.5 –4
–2.5 3
63)
17
Provide an appropriate response.
64)
Solve by the method of reduction:
2x – 9y= 10
x– 6y= 14
64)
65)
The price charged for cases of 2 different soft drinks at two different stores can be
represented by the matrix P=6 7
8 6
Store A
Store B . The number of cases of each soft drink sold
at each store can be represented by the matrix Q=100 80
60 90 . Show that the transpose of the
income generated (PQ)T is equal to the product of the transposes of P and Q in reverse
order, QTPT.
65)
66)
If B=
7 5 0
–2 1 1
1–3–5
and C=6 8
2–1, find BC.
66)
67)
The prices (in dollars per case) for 3 types of pens are represented by the price vector: P=
99 79 109 . An office supply store orders cases of these pens in the quantities given by
the column vector: Q=
5
3
6
. Find the total cost (in dollars) of the purchase.
67)
18
A women’s clothing chain takes inventory of one brand of sweater. The sweaters come in 3 sizes: small, medium, and
large, and 5 colors. The inventories at stores A, B, and C are represented by the matrices below.
Green
Blue
Black
Pink
Gold
A =
4 2 8
6 0 7
9 1 3
0 3 2
5 0 7
; B =
3 8 2
7 0 1
3 4 2
6 8 5
0 2 4
; C =
5 1 0
3 5 9
7 4 1
0 7 6
3 2 0
68)
Use the matrix operations on a graphing calculator to show that A+C=C+A.
68)
The energy economy of an area is composed of four industries: gas, coal, hydroelectric, and nuclear. The three main
consumers of energy are area residential consumers, a manufacturing plant, and a university. Assume that each consumer
may use some of the energy produced by each industry, and also that each industry uses some of the energy produced by
each of the other industries. The energy needs of each consumer and industry is represented by a demand vector whose
entries, in order, give the amount of gas, coal, hydroelectric, and nuclear energy needed by each consumer or industry, in
some convenient units. The demand vectors for the consumers are: DR=2 4 0 1 ; DM=3 0 2 5 ; DU=0 3 5 1 ;
and the demand vectors for the industries are: DG=0 1 3 2 ; DC=4 0 1 2 ; DH=3 1 0 1 ; DN=2 1 3 0 . The
price of gas is $15,000 per unit, the price of coal is $10,000 per unit, the price of hydroelectric power is $7000 per unit, and
the price of nuclear energy is $9000 per unit. These prices can be represented by the (column) price vector:
P =
15,000
10,000
7000
9000
69)
Find the income earned by the gas industry and its cost for the other forms of energy it
uses. Then calculate its profit.
69)
70)
Use the matrix operations on a graphing calculator to find the income earned by the
hydroelectric industry and its cost for the other forms of energy it uses. Then calculate its
profit.
70)
Provide an appropriate response.
71)
Let A=
1 2 3
2 5 7
–3–6–10
. Find A–1.
71)
72)
The price charged for 2 different paperback books at two different bookstores can be
represented by the matrix P=6 4
7 6
Store A
Store B . The quantities of each book sold at each store
can be represented by the matrix Q=48 35
26 18 . Use the matrix operations on a graphing
calculator to show that the transpose of the income generated (PQ)T is equal to the product
of the transposes of P and Q in reverse order, QTPT.
72)
73)
Find all solutions by reducing the matrix:
x–y–3z=2
x+y–z=1
2x–y–5z=7
2
73)
74)
Use matrix multiplication to represent the system:
4x+6z=3
x–y+z=4
3x+y–z=5
74)
20