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
79)
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.
79)
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.
80)
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.
80)
Provide an appropriate response.
81)
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.
81)
21
82)
An office furniture company manufactures desks and tables at two plants, A and B. Matrix
J represents the production of the two plants in January, and matrix F represents the
production of the two plants in February. Use the matrix operations on a graphing
calculator to write a matrix that represents the total product at the two plants for the two
months.
Desks
Tables J =120 80
105 130 ; F=110 140
85 125
82)
83)
Solve by the method of reduction:
2x – 5y= 10
3x–y= 2
83)
If E =1 3
2 4 is the encoding matrix, compute the following.
84)
E4(Use the matrix operations on a graphing calculator.)
84)
Provide an appropriate response.
85)
If A=
–8 3
2 1
1–7
and B=
5 2
–2 9
4–3
, find 2A– 3B.
85)
If E =1 3
2 4 is the encoding matrix, compute the following.
86)
E3
86)
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
87)
A researcher is studying the effects on the local economy of changes in the original demand
matrix. He models one change by the matrix equation D1=D+ 0.5D. Calculate the new
demand matrix in his model.
87)
Provide an appropriate response.
88)
If A=
1 0 3
–1 2 1
0 1 –1
, use row reduction to determine A–1 providing it exists.
88)
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
89)
Find the income earned by the coal industry and its cost for the other forms of energy it
uses. Then calculate its profit.
89)
90)
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
90)
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
91)
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.
91)
92)
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
92)
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 ].
93)
Use the matrix operations on a graphing calculator to find the total demand for the four
types of energy from the manufacturing facility and the university.
93)
94)
Reduce the matrix: 2 1 –1
4 0 1
94)
A manufacturer of doors, windows, and cabinets writes her yearly profit (in thousands of dollars) for each category in a
vector as: P =
248
319
532
. Her fixed costs of production can be described by the vector: C =
40
30
60
.
95)
Use the matrix operations on a graphing calculator to find her income if increasing it by
20% would cause an increase in profit of 50%. (Round to the nearest thousand dollars.)
95)
Provide an appropriate response.
96)
Perform the indicated operation and simplify your answer: [6][5]
96)
97)
Perform the indicated operation and simplify your answer: 1–1 0
2 3 4
1 0
5 1
–2 3
97)
98)
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
98)
Provide an appropriate response.
99)
If A=1 2 3
3 4 5 ; B=2–1–2
1–3–2, then find (A+B)T.
99)
100)
Use the matrix operations on a graphing calculator to find the transpose of
1 2 4 8 16
1 2 4 8 16
1 2 4 8 16
100)
101)
Solve the matrix equation: x
1
2
3
+ 2
3
5
1
+y
0
3
0
=
4
3
x–y– 3
101)
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.
102)
5 7
6 8 and –43.5
3–2.5
102)
5 7
Provide an appropriate response.
103)
Write A=aij if A is 2 × 2 and aij = 2i+j.
103)
104)
Find the inverse matrix of 2 1
–2 3 .
104)
26
105)
(a) If A is the coefficient matrix of the system x+3y=2
x+2y=5, determine A–1.
(b) Use A–1 to solve the system.
105)
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
.
106)
Use the matrix operations on a graphing calculator to find the profit made by the
manufacturer on these orders.
106)
Provide an appropriate response.
107)
Find the inverse matrix of 1 2
–3 4 .
107)
2
5–1
5
108)
A group of investors has $500,000 to invest in the stocks of three companies. Company A
sells for $50 a share and has an expected growth of 13% per year. Company B sells for $20
per share and has an expected growth of 15% per year. Company C sells for $80 a share
and has an expected growth of 10% per year. The group decides to try a new investment
strategy which entails buying equal amounts of shares in Company B and Company C, and
having a goal of 11.4% growth per year. How many shares of each stock should they buy?
108)
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
109)
A boom in the economy causes an across the board increase of 10% in energy demands in
the area. Calculate a new demand matrix which reflects this increase.
109)
Provide an appropriate response.
110)
Find a 3 × 2 matrix aij with a11 =a22 = 0, all other aij equal to 1.
110)
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
111)
A client wants 33 blocks of high–risk stock, 27 blocks of moderate–risk stock, and 30 blocks
of low–risk stock. How many of each portfolio should be suggested.
111)
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
112)
Use the matrix operations on a graphing calculator to show that A+ (B+C) = (A+B) +C.
112)
28
A health spa customizes the diet and vitamin supplements for each client. They offer three different vitamin
supplements, each containing different percentages of the recommended daily allowance (RDA) of vitamins A, C, and D.
One tablet of supplement X provides 40% of the RDA of A, 20% of the RDA of C, and 10% of the RDA of D. One tablet of
supplement Y provides 10% of the RDA of A, 10% of the RDA of C, and 30% of the RDA of D. One tablet of supplement
Z provides 10% of the RDA of A, 50% of the RDA of C, and 20% of the RDA of D.
113)
The spa staff determines that a client should take 140% of the RDA of vitamin A, 140% of
the RDA of vitamin C, and 190% of the RDA of vitamin D each day. How many tablets of
each supplement should he take each day?
113)
Provide an appropriate response.
114)
A manufacturer who requires raw materials A, B, C, D, and E is interested in tracking the
costs of these materials from 3 different sources. What is the order of the matrix he would
use?
114)
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 ].
115)
What is the total demand for the four types of energy from both the consumers and the
energy industries?
115)
Provide an appropriate response.
116)
Solve the following system of equations by reducing the matrix:
x–y–3z=2
2x–y–4z=3
x+y–z=1
116)
117)
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 BA – C.
117)
29
118)
Perform the indicated operations and simplify your answer: 3 0 0
–1 2
– 4 1 9
0–3
118)
A health spa customizes the diet and vitamin supplements for each client. They offer three different vitamin
supplements, each containing different percentages of the recommended daily allowance (RDA) of vitamins A, C, and D.
One tablet of supplement X provides 40% of the RDA of A, 20% of the RDA of C, and 10% of the RDA of D. One tablet of
supplement Y provides 10% of the RDA of A, 10% of the RDA of C, and 30% of the RDA of D. One tablet of supplement
Z provides 10% of the RDA of A, 50% of the RDA of C, and 20% of the RDA of D.
119)
The spa staff determines that a client should take 90% of the RDA of vitamin A, 190% of
the RDA of vitamin C, and 130% of the RDA of vitamin D each day. Use the matrix
operations on a graphing calculator to determine how many tablets of each supplement
should she take each day.
119)
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 ].
120)
Use the matrix operations on a graphing calculator to find the total demand for the four
types of energy from only the consumers.
120)
Provide an appropriate response.
121)
An employer offers medical, dental, and life insurance to both salaried and unsalaried
employees. She describes her annual costs for these benefits in a matrix, and finds that,
after employee contributions, only medical insurance requires an employer contribution of
$20,000 for salaried and $15,000 for unsalaried employees. Construct a matrix which shows
121)
30
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.
122)
If the letters to be encoded are L=b
e, show that (E + F)L= EL +FL.
122)
Provide an appropriate response.
123)
Find all solutions by reducing the matrix:
x–y–3z=2
x+y–z=1
2x–y–5z=7
2
123)
124)
Perform the indicated operation and simplify your answer: 2
43 1 0
124)
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.
125)
EO
125)
Provide an appropriate response.
126)
A blended juice product requires 5 gallons of apple juice, no orange juice, and 2 gallons of
cranberry juice. Construct an ingredient matrix for this product.
126)
127)
For what values of a will the following system of equations have a solution?
x–y–3z=2
x+y–z=1
2x–y–5z=a
127)
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
128)
Show that A+B=B+A.
128)
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
129)
On Saturdays, the company must move 23 tons of mail, 16 tons of medical supplies, and 17
tons of freight. This system of equations can be represented by the matrix:
2 4 1 23
1 2 1 16
3 2 0 17
.
Reduce this matrix to calculate how many aircraft of each type should be used.
129)
Provide an appropriate response.
130)
Solve the matrix equation: xy – 1
5 2x
=2y4
5 2x
130)
131)
The prices (in dollars per unit) for 3 video tapes are represented by the price vector:
P=24.95 16.95 18.95 . A video rental store orders these tapes in the quantities given by
the column vector: Q=
20
15
10
. Find the total cost (in dollars) of the purchase.
131)
132)
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.
132)
133)
If A=
–8 3
2 1
1–7
, B=
5 2
–2 9
4–3
, and C=
7–1
5–2
3–3
, find 4A– 2B + 3C.
133)
134)
Find the transpose of 5 0 4
9 5 3 .
134)
33
135)
Solve the matrix equation: 3
x
y
z
– 2
1
–2
3
=
1
10
3
135)
136)
Perform the indicated operations and simplify your answer:
3–1
4 2
6–8
+ 2
4–1
0 5
–4 3
136)
137)
If A=
5–2
–4 3
1 4
and C=6 8
2–1, find AC.
137)
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
.
138)
Use the matrix operations on a graphing calculator to find the cost of each type of vehicle.
138)
34
Provide an appropriate response.
139)
If A=
1 6 0 4
2 3 1 0
4 7 2 8
, determine (a) a32, and (b) the order of A.
139)
140)
Let A=1 2
3 4 ; find A–1.
140)
141)
Solve by the method of reduction:
2x – 9y= 10
x– 6y= 14
141)
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
142)
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
142)
Provide an appropriate response.
143)
Using the method of reduction, solve the system:
2x–y–4z=0
4x+y–2z=0
x–y–3z=0
143)
35
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
.
144)
Find the total cost of raw materials for these vehicles.
144)
Provide an appropriate response.
145)
The prices (in dollars per case) for 3 types of frozen pizzas are represented by the price
vector: P=75 52 48 . A grocery store orders cases of these pizzas in the quantities given
by the column vector: Q=
14
11
17
. Use the matrix operations on a graphing calculator to find
the total cost (in dollars) of the purchase.
145)
146)
Which of the following are reduced matrices?
A=0
B=0 1
0 0
C=
1 2 3
0 0 1
1 0 0
D=0 0 1 0
0 1 0 0
146)
36
147)
An electronics store has 35 televisions, 15 VCRs, and 25 CD players in stock. If the value of
each television is $400, each VCR is $200 and each CD player is $150, use the matrix
operations on a graphing calculator to find the total value of the electronics store’s
inventory.
147)
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
148)
Use the matrix operations on a graphing calculator to show that A+C=C+A.
148)
Provide an appropriate response.
149)
Reduce the matrix: 1 2 0
3–5 2
149)
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
150)
Show that A+O=A.
150)
37