75)
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?
75)
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
76)
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.
76)
Provide an appropriate response.
77)
If A=
–8 3
2 1
1–7
and B=
5 2
–2 9
4–3
, find A + B.
77)
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
78)
What is the change in sales between 1994 and 1996?
78)
21
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
79)
A client wants 26 blocks of high–risk stock, 25 blocks of moderate–risk stock, and 29 blocks
of low–risk stock. Use the matrix operations on a graphing calculator to find how many of
each portfolio should be suggested.
79)
Provide an appropriate response.
80)
A manufacturer who produces products A, B, and C is only interested in tracking labor
costs. What is the order of the matrix he would use?
80)
81)
Find transpose of
9–2–7
–5 5 3
4 6 –3
.
81)
82)
The prices (in dollars per unit) for 3 textbooks are represented by the price vector:
P=26.25 34.75 28.50 . A university bookstore orders these books in the quantities given
by the column vector: Q=
250
325
175
. Use the matrix operations on a graphing calculator to
find the total cost (in dollars) of the purchase.
82)
83)
Solve by the method of reduction:
x + 2y– 3z= 4
2x – 3y– 5z= – 1
–3x + 8y– 13z= 4
83)
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.
84)
Determine whether the intersection of the planes:
5x+2y+z=0
2x–y+3z=0
3x–2y+6z=0
has a unique solution or infinitely many solutions; then solve the system.
84)
Provide an appropriate response.
85)
Perform the indicated operation and simplify your answer: [6][5]
85)
86)
A pet store has 6 kittens, 10 puppies, and 7 parrots in stock, and a second store with 8
kittens, 14 puppies, and 9 parrots in stock. If the value of each kitten is $55, each puppy is
$150 and each parrot is $35, find the total value of the pet store’s inventory.
86)
87)
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.
87)
A zoo veterinarian can purchase animal food of 4 different types, A, B, C, and D. Each food comes in the same size bag,
and the number of grams of each of three nutrients in each bag are summarized in the following table. Use this
information to answer the questions that follow.
Food
A B C D
N15 5 10 5
Nutrient: N210 5 30 10
N3515 10 25
88)
For one animal, she determines that she needs to combine these bags to get 6000 g of N1,
12,000 g of N2, and 11,000 g of N3. How many bags of each type of food should she order?
88)
Provide an appropriate response.
89)
Give an example of 2 matrices A and B where A×BB×A.
89)
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.
90)
I–E
90)
Provide an appropriate response.
91)
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
91)
Set up a matrix equation with integer values that is equivalent to the system of equations.
92)
y=1
2x–3
10
y= – 3x+13
2
92)
24
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
93)
A new EPA mandate requires that all energy users reduce their energy use from the
original level by 1
3 over the next 10 years to minimize global warming. Use the matrix
operations on a graphing calculator to calculate a new demand matrix which reflects this
decrease.
93)
Provide an appropriate response.
94)
Solve by the method of reduction:
2x – 3y= – 4
–8x + 12y= 16
94)
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
95)
Show that A+O=A.
95)
96)
Show that A+B=B+A.
96)
25
Provide an appropriate response.
97)
If A=
5–2
–4 3
1 4
and B=
7 5 0
–2 1 1
1–3–5
, find BA.
97)
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
98)
What is the change in sales between 1994 and 1995?
98)
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.
99)
4(E+ 3I)
99)
Provide an appropriate response.
100)
Perform the indicated operations and simplify your answer: 2 3
1
–1 2
–3 0
1
0
100)
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
.
101)
Use the matrix operations on a graphing calculator to find her income if increasing it two
times would cause her profit to be three times as large. (Round to the nearest thousand
dollars.)
101)
26
A zoo veterinarian can purchase animal food of 4 different types, A, B, C, and D. Each food comes in the same size bag,
and the number of grams of each of three nutrients in each bag are summarized in the following table. Use this
information to answer the questions that follow.
Food
A B C D
N15 5 10 5
Nutrient: N210 5 30 10
N3515 10 25
102)
For one animal, she determines that she needs to combine these bags to get 15,000 g of N1,
30,000 g of N2, and 19,000 g of N3. How many bags of each type of food should she order?
102)
Provide an appropriate response.
103)
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?
103)
Explanation:
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.
104)
5 7
6 8 and –43.5
3–2.5
104)
Explanation:
If E =1 3
2 4 is the encoding matrix, compute the following.
105)
E4(Use the matrix operations on a graphing calculator.)
105)
Explanation:
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.
106)
Determine whether the intersection of the planes:
3x–2y–z=0
x+3y–5z=0
2x–7y+3z=0
has a unique solution or infinitely many solutions; then solve the system.
106)
Explanation:
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
.
107)
Find the amount of each raw material the manufacturer will need to fill the orders.
107)
Provide an appropriate response.
108)
A manufacturer of blended juices will also bottle pure juices for one supermarket chain.
This week he bottles cherry, grape, and cranberry juice. Fill in the matrix for the gallons of
each juice needed to produce 1 gallon of juice for each of the 3 pure juices.
Cherry Grape Cranberry
Cherry
Grape
Cranberry
108)
109)
Solve the matrix equation: x y +2
x+y4
=x1
0 4
109)
110)
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.
110)
28
111)
Find a 3 × 2 matrix aij with a11 =a22 = 0, all other aij equal to 1.
111)
Explanation:
112)
Solve by the method of reduction:
3x – 5y= 2
18x– 30y= 14
112)
Explanation:
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)
Explanation:
114)
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.
114)
Explanation:
Provide an appropriate response.
115)
Reduce the matrix: 2 1 –1
4 0 1
115)
Explanation:
29
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.
116)
If the letters to be encoded are L=b
e, show that (E + F)L= EL +FL.
116)
Provide an appropriate response.
117)
Using the method of reduction, solve the system:
3x–2y+z= – 1
2x–y–z=5
2x+3z=4
117)
A zoo veterinarian can purchase animal food of 4 different types, A, B, C, and D. Each food comes in the same size bag,
and the number of grams of each of three nutrients in each bag are summarized in the following table. Use this
information to answer the questions that follow.
Food
A B C D
N15 5 10 5
Nutrient: N210 5 30 10
N3515 10 25
118)
For one animal, she determines that she needs to combine these bags to get 9000 g of N1,
19,000 g of N2, and 17,000 g of N3. Use the matrix operations on a graphing calculator to
determine how many bags of each type of food should she order.
118)
Provide an appropriate response.
119)
Three components A, B, and C, are used in the manufacturing of a product. For a
production run, 100, 150, and 200 units of A, B, and C are $2, $1, and $3, respectively.
Represent the total cost of A, B, and C by a matrix product.
119)
30
120)
Find the transpose of 5 0 4
9 5 3 .
120)
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 ].
121)
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.
121)
Provide an appropriate response.
122)
Let A=1 2
2–1. Find A2.
122)
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.
123)
If the letters to be encoded are L=h
e, show that (E + F)L= EL + FL.
123)
31
Provide an appropriate response.
124)
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.
124)
Set up a matrix equation with integer values that is equivalent to the system of equations.
125)
y=4
7x–3
7
y= – 8x– 2
125)
If E =1 3
2 4 is the encoding matrix, compute the following.
126)
E3
126)
Provide an appropriate response.
127)
Write A=aij if A is 2 × 2 and aij = 2i+j.
127)
128)
Using the method of reduction, solve the system:
2x–y–4z=0
4x+y–2z=0
x–y–3z=0
128)
129)
By only looking at the following equations, can you say if the equations have a unique
solution or infinitely many solutions?
x+ 2y= 0
2z+ 3y= 0
3x+ 5y= 0
5x+ 7y= 0
129)
130)
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.
130)
131)
Find the inverse matrix of
1–2 0
2 1 2
0 0 –1
.
131)
132)
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
132)
133)
If A=
1 6 0 4
2 3 1 0
4 7 2 8
, determine (a) a32, and (b) the order of A.
133)
134)
Perform the indicated operations and simplify your answer: 3 0 0
–1 2
– 4 1 9
0–3
134)
If E =1 3
2 4 is the encoding matrix, compute the following.
135)
E2
135)
33
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
136)
Find the income earned by the coal industry and its cost for the other forms of energy it
uses. Then calculate its profit.
136)
Provide an appropriate response.
137)
Use matrix multiplication to represent the system: 3x+ 2y=2
4x+y=5
137)
3 2
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.
138)
1 2
3 4 and –2 1
1.5 –0.5
138)
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
$13,000. This information can be written as the column price vector P =
16,000
21,000
13,000
.
139)
Use the matrix operations on a graphing calculator to find the profit made by the
manufacturer on these orders.
139)
Provide an appropriate response.
140)
Solve by the method of reduction:
3x– 2y + 5z= 7
x + y–z= 2
5x+ 3z= 11
140)
A manufacturer of doors, windows, and cabinets writes her yearly profit (in thousands of dollars) for each category in a
vector as: P =
248
319
532
. Her fixed costs of production can be described by the vector: C =
40
30
60
.
141)
Use the matrix operations on a graphing calculator to find her income if increasing it by
20% would cause an increase in profit of 50%. (Round to the nearest thousand dollars.)
141)
Provide an appropriate response.
142)
Perform the indicated operation and simplify your answer: 2
43 1 0
142)
35
143)
If A=
–8 3
2 1
1–7
, B=
5 2
–2 9
4–3
, and C=
7–1
5–2
3–3
, find 7A + 6B – 3C.
143)
144)
A courier service has an order to deliver four products to a factory. The table below gives
the specifications of the products.
A B C D
Volume(ft3) 5 2 9 17
Weight(lb) 20 10 40 70
Value($) 300 200 700 1100
If the company’s large van can carry 500 cu ft.; 2100 lb, and is insured for $34,000, how
many units of each product can be carried? Write the answer in terms of A, B, C, and D.
144)
145)
Perform the indicated operations and simplify your answer:
3–1
4 2
6–8
+ 2
4–1
0 5
–4 3
145)
146)
Find the inverse matrix of 1 2
–3 4 .
146)
If E =1 3
2 4 is the encoding matrix, compute the following.
147)
E5(Use the matrix operations on a graphing calculator.)
147)
Provide an appropriate response.
148)
By using matrix reduction, solve the system:
x1+2x2–3x3+ x4=1
2x1+x2+2x4=8
148)
149)
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
this result.
149)
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
150)
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.
150)
Provide an appropriate response.
151)
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.
151)
152)
(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.
152)
153)
Look at the equations 2x+y=5
7x+4y=7
(a) Set up these equations in the matrix form Ax= b
(b) Find A–1
(c) Using A–1, solve the equations.
153)