Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
If A=aij where A is 2 × 2 and aij =i– 2j, then matrix A is given by
1)
A)
2 1
3 4 .
B)
–1–3
0–2.
C)
–1 0
–3–2.
D)
–1–4
–1–3.
E)
2–1
3 0 .
2)
1–1 2
4
–1
0
2 3 +
1 0
0 1
–1–1
=
2)
A)
2–3
4 8
1 0
B)
912
C)
5
–7
D)
2 9 –3
E)
none of the above
3)
Consider the matrix
5–9–1
0 7 –1
0 0 3
, this matrix can be best described as a(n):
3)
A)
main diagonal
B)
diagonal matrix
C)
lower triangular matrix
D)
upper triangular matrix
4)
If
x+y–3z=5
x–3y+z= – 7
2x–y–3z=2
, then
4)
A)
x= 5, y= – 8, and z= 12.
B)
x= 2 – 4z, y= – 5 + 2z, and z= t.
C)
x= 3, y= – 1, and z= 4.
D)
x= 5 – 3z, y= 3 + 4z, and z=t.
E)
none of the above
5)
If R=
0–1 0
1 2 –1
1 1 0
, then R–1=
5)
A)
2–1 2
–1 1 3
4 2 –2
B)
1 0 –3
–1 2 4
0 1 6
C)
1 0 1
–1 0 0
–1–1 1
D)
3
2
5
2–3
2
–14
3
1
2
22
3–5
3
E)
none of the above
6)
21 0 –4
2–3–1
–4–2 0
3–5 1
=
6)
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
7)
If 3x–y
x y
=3 2
2x y, then y=
7)
A)
4.
B)
–2.
C)
2.
D)
0.
E)
–4.
8)
Reducing
2 2 4
1 1 2
1 0 1
gives
8)
A)
1 0 1
0 1 1
0 0 0
B)
1 0 0
0 1 0
0 0 1
C)
1 1 2
0 0 0
1 0 1
D)
1 0 1
0 1 2
0 0 0
E)
1 0 3
0 0 1
0 0 0
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= 4.
C)
x=1
4and y=1
2.
D)
x= 0 and y= 0.
E)
There are no values for x and y which satisfy the equation.
10)
If x1–x2–3x3–4x4=3
3x1+x2–x3+4x4=5, then
10)
A)
x1= – 1 +x4, x2= 3x4, x3= 6 + 2x4, x4=x4
B)
x1= 3 – 6x4, x2= 4 +x4, x3= – 2 + 3x4, x4=x4
C)
x1= 2 +x3, x2= – 1 – 2x3– 4x4, x3=x3, x4=x4
D)
x1= 3 – 6x4, x2= 4 –x4, x3= – 2 + 3x4, x4=x4
E)
x1= – 1 +x3–x4, x2= 4 + 2x3+x4, x3=x3, x4=x4
11)
If
x–2y–4z=4
2x+y+z=9
x+y–z=1
, then
11)
A)
x= 2z, y= 3 – 4z, and z=t
B)
x= 5, y= – 5
2, and z=3
2
C)
x=22
5, y=1
5, and z= 0
D)
x= 4 –13
2z, y=11
2–1
6z, and z= t
E)
none of the above
12)
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
12)
A)
3 × 3.
B)
3 × 5.
C)
4 × 5.
D)
2 × 4.
E)
none of the above
13)
1–1
3 4
0 2
5–3
=
13)
A)
0–2
15 –12
B)
4–1
0 6
C)
–2 8
3 8
D)
–5 5
20 –6
E)
6 8
–4–17
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
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
14)
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.
14)
5
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.
15)
4(E+ 3I)
15)
16)
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.
16)
Provide an appropriate response.
17)
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.
17)
18)
The spa staff determines that a client should take 190% of the RDA of vitamin A, 150% of
the RDA of vitamin C, and 120% of the RDA of vitamin D each day. How many tablets of
each supplement should he take each day?
18)
Provide an appropriate response.
19)
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.
19)
7
20)
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.
20)
21)
If B=
7 5 0
–2 1 1
1–3–5
and C=6 8
2–1, find BC.
21)
22)
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.
22)
Provide an appropriate response.
23)
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.
23)
24)
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
24)
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
25)
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.
25)
26)
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.)
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
N15 5 10 5
N3515 10 25
27)
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?
27)
28)
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?
28)
29)
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.
29)
T
30)
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.
30)
31)
Find the income earned by the gas industry and its cost for the other forms of energy it
uses. Then calculate its profit.
31)
32)
Solve by the method of reduction:
2x – 3y= – 4
–8x + 12y= 16
32)
33)
If A=
2 1
1–7
and B=
–2 9
4–3
, find A–B.
33)
–13 1
34)
Use matrix multiplication to represent the system:
4x+6z=3
x–y+z=4
3x+y–z=5
34)
11
35)
Find transpose of
9–2–7
–5 5 3
4 6 –3
.
35)
36)
(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.
36)
37)
What is the total demand for the four types of energy from only the energy industries?
37)
38)
Perform the indicated operation if possible:
4
8
0
2 6 5
4–1 3
38)
39)
Solve the following system of equations by reducing the matrix:
2x+y=11
9x+5y=7
39)
Set up a matrix equation with integer values that is equivalent to the system of equations.
40)
y= – 3
4x+ 3
y= – 1
5x+9
5
40)
41)
What would her income be if increasing it threefold would cause her profit to be five times
as large? (Round to the nearest thousand dollars.)
41)
Provide an appropriate response.
42)
Solve by the method of reduction:
3x– 2y + 5z= 7
x + y–z= 2
5x+ 3z= 11
42)
43)
If the letters to be encoded are L=h
e, show that (E + F)L= EL + FL.
43)
44)
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
44)
45)
Find x, y, z, u, t, v such that x y z
u t v
+ 2 1 2 3
4 5 6
=8 3 5
9 8 14
45)
46)
Let A=1 1
2–1; B=3–5
–9 2 ; C=–2 2
4–1. Find AB+ 2C
46)
47)
Solve the following system of equations by reduction.
2x+ 3y+ 4z+ 9u= 4
3x+ 4y+ 8z+ 7u= 1
47)
48)
Write A=aij if A is 2 × 3 and aij = 2i+j.
48)
49)
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.
49)
14
50)
Find the amount of each raw material the manufacturer will need to fill the orders.
50)
6 8 , to answer the question.
51)
If the letters to be encoded are L=n
o, show that E(FL) = (EF)L.
51)
Provide an appropriate response.
52)
Let A=1 2
2–1. Find A2.
52)
15
53)
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.
53)
Provide an appropriate response.
54)
Give an example of 2 matrices A and B where A×BB×A.
54)
55)
A bookstore has 100 dictionaries, 70 cookbooks, and 90 thesauruses in stock, and a second
store with 85 dictionaries, 95 cookbooks, and 60 thesauruses in stock. If the value of each
dictionary is $28, each cookbook is $22 and each thesaurus is $16, use the matrix operations
on a graphing calculator to find the total value of the bookstore’s inventory.
55)
100 85
56)
Find the transpose of the matrix:
1 3 3
4 5 6
7 8 9
56)
57)
EF
57)
Provide an appropriate response.
58)
Perform the indicated operations and simplify your answer: 2 3
1
–1 2
–3 0
1
0
58)
59)
Write a diagonal matrix of size 3 × 3 with entries aii =i+ 3, aij = 0 for ij.
59)
60)
If A=
–8 3
2 1
1–7
and B=
5 2
–2 9
4–3
, find –4A + 5B.
60)
Set up a matrix equation with integer values that is equivalent to the system of equations.
61)
y=4
7x–3
7
y= – 8x– 2
61)
Provide an appropriate response.
62)
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
62)
63)
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.
63)
64)
Solve by the method of reduction:
3x – 5y= 2
18x– 30y= 14
64)
65)
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.
65)
66)
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.
66)
67)
By looking at the solution of the above question, can you give the number of solutions of
the equations?
3x+ 5y– 11z= 0
9x+ 13y– 15z= 12
67)
68)
If A=
2 7 4
1 9 5
3 6 8
, determine (a) a23, and (b) the order of A.
68)
69)
Solve by the method of reduction:
5x + 3y + z= 2
2x–y+ 2z= 10
4x– 2y+ 3z= 17
69)
70)
What is the change in sales between 1994 and 1996?
70)
71)
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.
71)
Provide an appropriate response.
72)
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.
72)
73)
Find the income earned by the nuclear industry and its cost for the other forms of energy it
uses. Then calculate its profit.
73)
74)
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.
74)
75)
What is the change in sales between 1995 and 1996?
75)
Provide an appropriate response.
76)
If A=
–8 3
2 1
1–7
and B=
5 2
–2 9
4–3
, find A + B.
76)
77)
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.
77)
78)
6 8
5 7 and 3.5 –4
–2.5 3
78)
6 8
20
Provide an appropriate response.