154)
If A=
–8 3
2 1
1–7
and B=
5 2
–2 9
4–3
, find 2A– 3B.
154)
155)
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
155)
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
.
156)
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.)
156)
Provide an appropriate response.
157)
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
157)
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 ].
158)
What is the total demand for the four types of energy from both the consumers and the
energy industries?
158)
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.
159)
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?
159)
Provide an appropriate response.
160)
Perform the indicated operation and simplify your answer: 1–1 0
2 3 4
1 0
5 1
–2 3
160)
161)
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
161)
162)
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.
162)
163)
Solve the matrix equation: x
1
2
3
+ 2
3
5
1
+y
0
3
0
=
4
3
x–y– 3
163)
164)
If A=
5–2
–4 3
1 4
and C=6 8
2–1, find AC.
164)
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
165)
Find the income earned by the nuclear industry and its cost for the other forms of energy it
uses. Then calculate its profit.
165)
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
166)
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.
166)
Provide an appropriate response.
167)
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.
167)
40
168)
Solve the following system of equations by reducing the matrix:
x–y–3z=2
2x–y–4z=3
x+y–z=1
168)
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
169)
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.
169)
Provide an appropriate response.
170)
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
170)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
171)
Given the coefficient matrix A and the final demand matrix D stated below, find (I – A)–1 and find
the output matrix X.
A =0.3 0.3
0.2 0.2 C =70
30
171)
A)
(I – A)–1=0.6 1.6
0.4 1.4 ; X =130
70
B)
(I – A)–1=1.6 0.6
0.4 1.4 ; X =70
130
C)
(I – A)–1=1.6 0.6
0.4 1.4 ; X =130
70
D)
(I – A)–1=0.6 1.6
0.4 1.4 ; X =70
130
172)
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.
172)
A)
$52 billion agriculture, $44 billion manufacturing
B)
$44 billion agriculture, $52 billion manufacturing
C)
$56 billion agriculture, $36 billion manufacturing
D)
$36 billion agriculture, $56 billion manufacturing
173)
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 coefficient matrix that
would be used in determining the total output of each grade of gasoline.
173)
A)
A =
0.13 0.16 0.17
0.14 0.15 0.13
0.15 0.17 0.11
B)
A =
0.14 0.18 0.17
0.14 0.15 0.13
0.15 0.17 0.11
C)
A =
0.13 0.18 0.17
0.16 0.15 0.13
0.15 0.17 0.11
D)
A =
0.14 0.14 0.15
0.18 0.15 0.17
0.17 0.13 0.11
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