Deck 5: Matrix Algebra and Applications

Question 1

Four sectors of some economy are (1) Sector 1, (2) Sector 2, (3) Sector 3, and (4) Sector 4. The input-output table involving these four sectors was as follows (all figures are in millions of dollars): $$\begin{array} { | r | r | r | r | r | } \hline \text { To } &1 & 2 & 3 & 4 \\\hline \text { From } \mathbf { 1 } & 171 & 30.6 & 120.2 & 14.2 \\\hline \mathbf { 2 } & 0 & 190.1 & 55.7 & 12.6 \\\hline \mathbf { 3 } & 2.9 & 40.2 & 1,418.1 & 1,242 \\\hline \mathbf { 4 } & 0.1 & 7.2 & 40.3 & 326.4 \\\hline \text { Total Output } & 3,278.9 & 2,189 & 6,541.3 & 4,065.3 \\\hline\end{array}$$ How much additional production by the Sector 1 is necessary to accommodate a $1,000 increase in the demand for the products of Sector 4 Please round your answer to two decimal places.

A)$12.78 million
B) $12.38 million
C) $12.58 million
D) $12.28 million
E) $12.08 million

Answer 1

$12.38 million

Question 2

Given the technology matrix A, and an external demand vector D, find the production vector X.
$A = \left[ \begin{array} { l l } 0.4 & 0.3 \\0.1 & 0.2\end{array} \right]$ , $D = \left[ \begin{array} { l } 30,000 \\25,000\end{array} \right]$

A) $X = \left[ \begin{array} { l } 40,000 \\70,000\end{array} \right]$
B) $X = \left[ \begin{array} { l } 70,000 \\40,000\end{array} \right]$
C) $X = \left[ \begin{array} { l } 64,000 \\40,000\end{array} \right]$
D) $X = \left[ \begin{array} { l } 70,000 \\64,000\end{array} \right]$
E) $X = \left[ \begin{array} { l } 40,000 \\64,000\end{array} \right]$

Answer 2

$X = \left[ \begin{array} { l } 70,000 \\40,000\end{array} \right]$

Question 3

Each unit of television news requires 1.1 units of television news and 0.1 units of radio news. Each unit of radio news requires 0.3 units of television news and no radio news. With sector 1 as television news and sector 2 as radio news, set up the technology matrix A .

A) $A = \left[ \begin{array} { c c } 0.1 & 0.3 \\1.1 & 0\end{array} \right]$
B) $A = \left[ \begin{array} { c c } 1.1 & 0.3 \\0.1 & 0\end{array} \right]$
C) $A = \left[ \begin{array} { c c } 1.1 & 0.3 \\0.3 & 0\end{array} \right]$
D) $A = \left[ \begin{array} { l } 1.4 \\0.1\end{array} \right]$
E) $A = \left[ \begin{array} { l l } 0.3 & 0 \\0.1 & 1\end{array} \right]$

Answer 3

$A = \left[ \begin{array} { c c } 1.1 & 0.3 \\0.1 & 0\end{array} \right]$

Question 4

Two sectors of any country economy are (1) audio, video, and communication equipment and (2) electronic components and accessories. In 1998, the input-output table involving these two sectors was as follows (all figures are in millions of dollars): $$\begin{array} { | r | c | c | } \hline \text { To } & \text { Equipment } & \text { Components } \\\hline \text { From Equipment } & 8,000 & 500 \\\hline \text { Components } & 26,000 & 34,000 \\\hline \text { Total output } & 82,000 & 121,000 \\\hline\end{array}$$
Determine the production levels necessary in these two sectors to meet an external demand for $60,000 million of communication equipment and $90,000 million of electronic components. Round answers to two significant digits.

A)Equipment Sector production approximately $64,195.33 million, Components Sector production approximately $154,804.67 million.
B) Equipment Sector production approximately $67,195.33 million, Components Sector production approximately $154,804.67 million.
C) Equipment Sector production approximately $64,195.33 million, Components Sector production approximately $144,804.67 million.
D) Equipment Sector production approximately $67,195.33 million, Components Sector production approximately $144,804.67 million.
E) Equipment Sector production approximately $144,804.67 million, Components Sector production approximately $64,195.33 million.

Question 5

Two sectors of some economy are Sector 1 and Sector 2. The input-output table involving these two sectors results in the following value for $$( I - A ) ^ { - 1 }$$ $$( I - A ) ^ { - 1 } = \left[ \begin{array} { c c } 1.0965 & 0.04361 \\0.00536 & 1.07515\end{array} \right]$$
How many additional dollars worth of production of Sector 2 must be produced to meet a $1 increase in the demand for products of Sector 2

A)$0.00536
B) $0.54825
C) $1.07515
D) $1.0965
E) $0.04361

Question 6

Production of 1 unit of cologne requires 0.9 units of perfume and 0.4 units of cologne. Into 1 unit of perfume goes 0.2 unit of perfume and 0.5 units of cologne. With sector 1 as cologne and sector 2 as perfume, set up the technology matrix A.

A) $A = \left[ \begin{array} { l l } 0.2 & 0.4 \\0.9 & 0.5\end{array} \right]$
B) $A = \left[ \begin{array} { l l } 0.2 & 0.4 \\0.5 & 0.9\end{array} \right]$
C) $A = \left[ \begin{array} { c c } 0.4 & 0.5 \\0.9 & 0\end{array} \right]$
D) $A = \left[ \begin{array} { l l } 0.4 & 0.5 \\0.9 & 0.2\end{array} \right]$
E) $A = \left[ \begin{array} { l l } 0.5 & 0.4 \\0.9 & 0.2\end{array} \right]$

Question 7

Two sectors of any country economy are (1) lumber and wood products and (2) paper and allied products. In 1998 the input-output table involving these two sectors was as follows (all figures are in millions of dollars). $$\begin{array} { | r | c | c | } \hline \text { To } & \text { Wood } & \text { Paper } \\\hline \text { From Wood } & 36,000 & 7,000 \\\hline \text { Paper } & 100 & 17,000 \\\hline \text { Total Output } & 120,000 & 120,000 \\\hline\end{array}$$ If external demand for lumber and wood products rises by $14,000 million and external demand for paper and allied products rises by $22,000 million, what increase in output of these two sectors is necessary Round answers to two significant digits.

A)increase in output of lumber and wood products is $22,137.71 million and increase in output of paper and allied products is $25,652.56 million
B) increase in output of lumber and wood products is $24,137.71 million and increase in output of paper and allied products is $28,652.56 million
C) increase in output of lumber and wood products is $21,137.71 million and increase in output of paper and allied products is $25,652.56 million
D) increase in output of lumber and wood products is $19,137.71 million and increase in output of paper and allied products is $23,652.56 million
E) increase in output of lumber and wood products is $22,137.71 million and increase in output of paper and allied products is $23,652.56 million

Question 8

Let A be the technology matrix, where Sector 1 is computer chips, and Sector 2 is silicon. $A = \left[ \begin{array} { c c } 0.06 & 0.001 \\0.5 & 0.002\end{array} \right]$
How many units of computer chips are needed to produce one unit of silicon
How many units of silicon are needed to produce one unit of silicon
How many units of silicon are needed to produce one unit of computer chips

A) computer chips $\rightarrow$ silicon: $0.001$
silicon $\rightarrow$ silicon: $0.5$
silicon $\rightarrow$ computer chips: $0.002$

B) computer chips $\rightarrow$ silicon: $0.06$
silicon $\rightarrow$ silicon: $0.5$
silicon $\rightarrow$ computer chips: $0.002$

C) computer chips $\rightarrow$ silicon: $0.001$
silicon $\rightarrow$ silicon: $0.5$
silicon $\rightarrow$ computer chips: $0.5$

D) computer chips $\rightarrow$ silicon: $0.001$
silicon $\rightarrow$ silicon: $0.002$
silicon $\rightarrow$ computer chips: $0.5$

E) computer chips $\rightarrow$ silicon: $0.5$
silicon $\rightarrow$ silicon: $0.002$
silicon $\rightarrow$ computer chips: $0.001$

Question 9

Let $( I - A ) ^ { - 1 } = \left[ \begin{array} { c c c } 1.3 & 0.1 & 0 \\0.2 & 1.1 & 0.3 \\0 & 0 & 1.2\end{array} \right]$ and assume that the external demand for the products in each of the sectors increases by 1 unit. By how many units should each sector increase production

A) $\left[ \begin{array} { c } 1.2 \\1.8 \\1\end{array} \right]$
B) $\left[ \begin{array} { l } 1.2 \\1.9 \\0.9\end{array} \right]$
C) $\left[ \begin{array} { c } 1.8 \\2 \\0.8\end{array} \right]$
D) $\left[ \begin{array} { l } 1.5 \\1.5 \\1.1\end{array} \right]$
E) $\left[ \begin{array} { l } 1.4 \\1.6 \\1.2\end{array} \right]$

Question 10

Four sectors of some economy are (1) Sector 1, (2) Sector 2, (3) Sector 3, and (4) Sector 4. The input-output table involving these four sectors was as follows (all figures are in millions of dollars) $$\begin{array} { | r | r | r | r | r | } \hline \text { To } &1 &2 & 3& 4\\\hline \text { From } \mathbf { 1 } & 693.3 & 3.4 & 3,407.9 & 1,006.5 \\\hline \mathbf { 2 } & 12.5 & 5.9 & 17.9 & 111.1 \\\hline \mathbf { 3 } & 47.6 & 4.4 & 917.5 & 164.1 \\\hline \mathbf { 4 } & 349.7 & 21 & 77.2 & 704.9 \\\hline \text { Total Output } & 9.471 .5 & 676.3 & 6,755.5 & 4,653.8 \\\hline\end{array}$$ How much additional production by the Sector 3 is necessary to accommodate a $100 increase in the demand for the products of Sector 1

A)$1.14
B) $0.72
C) $0.96
D) $0.42
E) $0.83

Question 11

Let $( I - A ) ^ { - 1 } = \left[ \begin{array} { c c c } 1.6 & 0.1 & 0 \\0.2 & 1.1 & 0.2 \\0.1 & 0.4 & 1.2\end{array} \right]$ and assume that the external demand for the products in Sector 1 increases by 1 unit. By how many units should each sector increase production

A) $\left[ \begin{array} { c } 1.6 \\0.1 \\0\end{array} \right]$
B) $\left[ \begin{array} { c } 0 \\1.1 \\0.1\end{array} \right]$
C) $\left[ \begin{array} { l } 1.6 \\0.2 \\1.2\end{array} \right]$
D) $\left[ \begin{array} { l } 1.6 \\0.2 \\0.1\end{array} \right]$
E) $\left[ \begin{array} { l } 1.6 \\1.1 \\1.2\end{array} \right]$

Question 12

Let A be the technology matrix, where Sector 1 is paper, and Sector 2 is wood. $$A = \left[ \begin{array} { l l } 0.6 & 0.05 \\0.8 & 0.04\end{array} \right]$$
How many units of wood are needed to produce one unit of paper
How many units of paper are needed to produce one unit of paper
How many units of paper are needed to produce one unit of wood

A) Wood $\rightarrow$ Paper: $0.8$
Paper $\rightarrow$ Paper: $0.6$
Paper $\rightarrow$ Wood: $0.05$

B) Wood $\rightarrow$ Paper: $0.04$
Paper $\rightarrow$ Paper: $0.6$
Paper $\rightarrow$ Wood: $0.05$

C) Wood $\rightarrow$ Paper: $0.04$
Paper $\rightarrow$ Paper: $0.6$
Paper $\rightarrow$ Wood: $0.05$ .

D) Wood $\rightarrow$ Paper: $0.04$
Paper $\rightarrow$ Paper: $0.04$
Paper $\rightarrow$ Wood: $0.8$

E) Wood $\rightarrow$ Paper: $0.04$
Paper $\rightarrow$ Paper: $0.05$
Paper $\rightarrow$ Wood: $0.6$

Question 13

Two student groups at Enormous State University, the Choral Society and the Football Club, maintain files of term papers that they write and offer to students for research purposes. Some of these papers they use themselves in generating more papers. To avoid suspicion of plagiarism by faculty members (who seem to have astute memories), each paper is given to students or used by the clubs only once (no copies are kept). The number of papers that were used in the production of new papers last year is shown in the input-output table below. $$\begin{array} { | r | c | c | } \hline \text { To } & \text { Choral Society } & \text { Football Club } \\\hline \text { From } \text { Choral Society } & 20 & 35 \\\hline \text { Football Club } & 35 & 30 \\\hline \text { Total Output } & 100 & 200 \\\hline\end{array}$$ Given that 990 Choral Society papers and 1,485 Football Club papers will be used by students outside of these two clubs next year, how many new papers do the two clubs need to write

A)Choral Society 1,780, Football Club 2,480
B) Choral Society 1,730, Football Club 2,524
C) Choral Society 1,812, Football Club 2,466
D) Choral Society 1,770, Football Club 2,450
E) Choral Society 2,500, Football Club 1,830

Question 14

Given the technology matrix A, and an external demand vector D, find the production vector X. Round your answer to one decimal places. $A = \left[ \begin{array} { c c c } 0.2 & 0.3 & 0 \\0 & 0.2 & 0.3 \\0 & 0 & 0.2\end{array} \right]$ , $D = \left[ \begin{array} { l } 2000 \\4800 \\3300\end{array} \right]$

A) $X = \left[ \begin{array} { c } 6,000.0 \\5,333.3 \\32,222.2\end{array} \right]$
B) $X = \left[ \begin{array} { c } 32,222.2 \\6,000.0 \\5,333.3\end{array} \right]$
C) $X = \left[ \begin{array} { c } 32,222.2 \\5,333.3 \\6,000.0\end{array} \right]$
D) $X = \left[ \begin{array} { c } 6,000.0 \\32,222.2 \\5,333.3\end{array} \right]$
E) $X = \left[ \begin{array} { l } 5201.2 \\8204.1 \\4714.3\end{array} \right]$

Question 15

Given the technology matrix A, and an external demand vector D, find the production vector X. $A = \left[ \begin{array} { c c c } 0.25 & 0.25 & 0.25 \\0.25 & 0.7 & 0.25 \\0.25 & 0.25 & 0.25\end{array} \right] , D = \left[ \begin{array} { c } 9,000 \\18,000 \\9,000\end{array} \right]$

A) $X = \left[ \begin{array} { l } 540,000 \\288,000 \\288,000\end{array} \right]$
B) $X = \left[ \begin{array} { l } 288,000 \\288,000 \\540,000\end{array} \right]$
C) $X = \left[ \begin{array} { l } 345,600 \\432,000 \\345,000\end{array} \right]$
D) $X = \left[ \begin{array} { l } 288,000 \\540,000 \\288,000\end{array} \right]$
E) $X = \left[ \begin{array} { l } 345,600 \\432,000 \\345,600\end{array} \right]$

Question 16

Four sectors of some economy are (1) Sector 1, (2) Sector 2, (3) Sector 3, and (4) Sector 4. The input-output table involving these four sectors was as follows (all figures are in millions of dollars) $\begin{array}{|r|c|c|c|c|}\hline \text { To } & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} \\\hline \text{from }\mathbf{ 1} & 11,534 & 8 & 100 & 868 \\\hline \mathbf{2} & 23,307 & 4,159 & 0 & 4,931 \\\hline \mathbf{3} & 0 & 0 & 403 & 66 \\\hline \mathbf{4} & 5,281 & 10,934 & 3,285 & 223 \\\hline \text { Total Output } & 99,840 & 116,189 & 13,786 & 45,799 \\\hline\end{array}$ Determine how these four sectors would react to an increase in demand for Sector 1 production of $1,000 million.

A) $\left[ \begin{array} { c } 1,407.64 \\252.13 \\0.14 \\47.69\end{array} \right]$
B) $\left[ \begin{array} { c } 1,132.50 \\283.91 \\0.13 \\87.08\end{array} \right]$
C) $\left[ \begin{array} { c } 1,132.53 \\254.07 \\0.16 \\47.69\end{array} \right]$
D) $\left[ \begin{array} { c } 1,238.23 \\299.83 \\0.10 \\139.60\end{array} \right]$
E) $\left[ \begin{array} { c } 1,270.07 \\315.75 \\0.10 \\100.21\end{array} \right]$

Question 17

Four sectors of some economy are (1) Sector 1, (2) Sector 2, (3) Sector 3, and (4) Sector 4. The input-output table involving these four sectors was as follows (all figures are in millions of dollars) $\begin{array}{|r|r|r|r|r|}\hline \text { To } & 1 &2& \mathbf{3} & \mathbf{4} \\\hline \text{from }\mathbf{1} & 79 & 1,094 & 0 & 1,233 \\\hline \mathbf{2} & 64,859 & 13,086 & 7 & 1,074 \\\hline \mathbf{3} & 0 & 0 & 21,787 & 0 \\\hline \mathbf{4} & 0 & 0 & 0 & 1,375 \\\hline \text { Total Output } & 230,677 & 135,110 & 129,374 & 44,140 \\\hline\end{array}$ Determine how these four sectors would react to an increase in demand for Sector 1 production of $1000 million.

A) $\left[ \begin{array} { c } 1,140.44 \\299.63 \\0.01 \\13.13\end{array} \right]$
B) $\left[ \begin{array} { c } 1,278.01 \\324.77 \\- 0.01 \\- 39.39\end{array} \right]$
C) $\left[ \begin{array} { c } 1,153.02 \\305.92 \\0.01 \\52.52\end{array} \right]$
D) $\left. \begin{array} { c } 1,002.87 \\312.21 \\0 \\0\end{array} \right]$
E) $\left[ \begin{array} { c } 1,153.02 \\305.92 \\0.01 \\0\end{array} \right]$

Question 18

Two sectors of some economy are Sector 1 and Sector 2. The input-output table involving these two sectors results in the following value for $$( I - A ) ^ { - 1 }$$ $$( I - A ) ^ { - 1 } = \left[ \begin{array} { c c } 1.21 & 0.155 \\0.002 & 1.1242\end{array} \right]$$
How many additional dollars worth of production of Sector 2 must be produced to meet a $1 increase in the demand for products of Sector 1

A)$1.1242
B) $0.9692
C) $0.002
D) $1.21
E) $0.155

Question 19

Given the technology matrix A, and an external demand vector D, find the production vector X.
$A = \left[ \begin{array} { c c } 0.2 & 0.3 \\0 & 0.4\end{array} \right]$ , $D = \left[ \begin{array} { l } 14,000 \\15,000\end{array} \right]$

A) $X = \left[ \begin{array} { l } 26,875 \\25,000\end{array} \right]$
B) $X = \left[ \begin{array} { l } 25,000 \\26,875\end{array} \right]$
C) $X = \left[ \begin{array} { l } 53,750 \\26,875\end{array} \right]$
D) $X = \left[ \begin{array} { l } 25,000 \\53,750\end{array} \right]$
E) $X = \left[ \begin{array} { l } 53,750 \\53,750\end{array} \right]$

Question 20

Obtain the technology matrix from the input-output table. $$\begin{array} { | r | c | c | c | } \hline \text { to } & \text { A } & \text { B } & \text { C } \\\hline \text { from } \mathbf { A } & 0 & 200 & 300 \\\hline \mathbf { B } & 500 & 400 & 300 \\\hline \mathbf { C } & 0 & 0 & 600 \\\hline \text { Total Output } & 1,000 & 2,000 & 3,000 \\\hline\end{array}$$

A) $A = \left[ \begin{array} { c c c } 0.2 & 0.5 & 0 \\0.1 & 0 & 0 \\0.1 & 0.1 & 0.2\end{array} \right]$
B) $A = \left[ \begin{array} { c c c } 0 & 0 & 0.5 \\0.1 & 0.2 & 0 \\0.1 & 0.1 & 0.2\end{array} \right]$
C) $A = \left[ \begin{array} { c c c } 0 & 0.1 & 0.1 \\0.5 & 0.2 & 0.1 \\0 & 0 & 0.2\end{array} \right]$
D) $A = \left[ \begin{array} { c c c } 0 & 0.5 & 0 \\0.1 & 0.2 & 0 \\0.1 & 0.1 & 0.2\end{array} \right]$
E) $A = \left[ \begin{array} { c c c } 0.2 & 0 & 0.5 \\0.2 & 0 & 0 \\0.1 & 0.1 & 0.1\end{array} \right]$

Question 21

Given the technology matrix A, and an external demand vector D, find the production vector X.
​
​ ,

Question 22

Let A be the technology matrix, where Sector 1 = wood, and Sector 2 = paper.
​ ​
__________ units of paper are needed to produce one unit of wood.
​
__________ units of wood are needed to produce one unit of paper.
​
The production of each unit of paper requires the use of __________ units of paper.

Question 23

Given the technology matrix A, and an external demand vector B, find the production vector X.
​
​ ,

Question 24

Let $( I - A ) ^ { - 1 } = \left[ \begin{array} { c c c } 1.5 & 0.1 & 0 \\0.2 & 1.2 & 0.2 \\0.1 & 0.7 & 1.6\end{array} \right]$ and assume that the external demand for the products in Sector 1 increases by 1 unit. By how many units should each sector increase production
?
Express the answer as a column matrix.

Question 25

Each unit of television news requires 0.5 units of television news and 0.4 units of radio news. Each unit of radio news requires 0.7 units of television news and no radio news. With sector 1 as television news and sector 2 as radio news, set up the technology matrix A.

Question 26

Production of 1 unit of cologne requires 0.7 units of perfume and 0.3 units of cologne. Into 1 unit of perfume goes 0.2 unit of perfume and 0.4 units of cologne. With sector 1 as cologne and sector 2 as perfume, set up the technology matrix A.

Question 27

Given the technology matrix A, and an external demand vector D, find the production vector X.
​ ,

Question 28

Two student groups at Enormous State University, the Choral Society and the Football Club, maintain files of term papers that they write and offer to students for research purposes. Some of these papers they use themselves in generating more papers. To avoid suspicion of plagiarism by faculty members (who seem to have astute memories), each paper is given to students or used by the clubs only once (no copies are kept). The number of papers that were used in the production of new papers last year is shown in the input-output table below. $$\begin{array} { | r | r | r | } \hline \text { To } & \text { Choral Society } & \text { Football Club } \\\hline \text { From Choral Society } & 40 & 10 \\\hline \text { Football Club } & 10 & 15 \\\hline \text { Total Output } & 100 & 200 \\\hline\end{array}$$
Given that 110 Choral Society papers and 330 Football Club papers will be used by students outside of these two clubs next year, how many new papers do the two clubs need to write
Choral Society __________, Football Club __________

Question 29

Two sectors of some economy are Sector 1 and Sector 2. The input-output table involving these two sectors results in the following value for $$( I - A ) ^ { - 1 }$$ $$( I - A ) ^ { - 1 } = \left[ \begin{array} { l l } 1.313 & 0.162 \\0.007 & 1.1331\end{array} \right]$$
How many additional dollars worth of production of Sector 2 must be produced to meet a $1 increase in the demand for products of Sector 1 Round your answer to three decimal places.

Question 30

Given the technology matrix A, and an external demand vector D, find the production vector X.
​ ,

Question 31

Four sectors of some economy are (1) Sector 1, (2) Sector 2, (3) Sector 3, and (4) Sector 4. The input-output table involving these four sectors was as follows (all figures are in millions of dollars) $$\begin{array} { | r | r | r | r | r | } \hline \text { To } & 1& 2 & 3 & 4 \\\hline \text { From } \mathbf { 1 } & 681.8 & \mathbf { 4 } & 3,106.9 & 1,033.5 \\\hline \mathbf { 2 } & 12.8 & 7.2 & 15 & 103.4 \\\hline \mathbf { 3 } & 48.3 & 4.5 & 855.9 & 112.3 \\\hline \mathbf { 4 } & 348.7 & 22.5 & 82.1 & 693.7 \\\hline \text { Total Output } & 9,375.7 & 663.2 & 6,510.3 & 4,865.8 \\\hline\end{array}$$
How much additional production by the Sector 3 is necessary to accommodate a $100 increase in the demand for the products of Sector 1 Round your answers to two decimal places.
$ __________

Question 32

Two sectors of some economy are Sector 1 and Sector 2. The input-output table involving these two sectors results in the following value for $$( I - A ) ^ { - 1 }$$ $$( I - A ) ^ { - 1 } = \left[ \begin{array} { c c } 1.0519 & 0.07824 \\0.00379 & 1.09831\end{array} \right]$$
How many additional dollars worth of production of Sector 2 must be produced to meet a $1 increase in the demand for products of Sector 2 Round your answer to four decimal places.

Question 33

Four sectors of some economy are (1) Sector 1, (2) Sector 2, (3) Sector 3, and (4) Sector 4. The input-output table involving these four sectors was as follows (all figures are in millions of dollars)

Question 34

Obtain the technology matrix from the following input-output table. ​

Question 35

Let $( I - A ) ^ { - 1 } = \left[ \begin{array} { c c c } 1.3 & 0.2 & 0 \\0.3 & 1.2 & 0.3 \\0 & 0 & 1.1\end{array} \right]$ and assume that the external demand for the products in each of the sectors increases by 1 unit. By how many units should each sector increase production

Question 36

Reduce the payoff matrix by dominance.
$$\begin{array} { l } \quad\quad\quad\quad\quad\text { B } \\\begin{array} { l l l } \quad\quad\quad\quad p & q & r\end{array} \\A \begin{array} { l } a \\b\end{array} \left[ \begin{array} { c c c } 2 & 0 & 10 \\13 & - 2 & - 7\end{array} \right] \\\end{array}$$

A) $\begin{array} { l } \quad\quad\quad\quad\text { B } \\\quad\quad\quad\quad p \quad q \\\mathrm { A } \begin{array} { l } a \\{ b }\end{array} \left[ \begin{array} { l l } 2 & 13 \\0 & - 2\end{array} \right] \\\end{array}$
B) $\begin{array} { l } \quad\quad\quad\quad\text { B } \\\quad\quad\quad\quad p \quad q \\\mathrm { A } \begin{array} { l } a \\ { b }\end{array} \left[ \begin{array} { l l } 2 & 0 \\13&-2\end{array} \right] \\\end{array}$
C) $\begin{array} { l } \quad\quad\quad\quad\text { B } \\\quad\quad\quad\quad p \quad r \\\mathrm { A } \begin{array} { l } a \\ { b }\end{array} \left[ \begin{array} { l l } 0&10 \\-2&-7\end{array} \right] \\\end{array}$
D) $\begin{array} { l } \quad\quad\quad\quad\text { B } \\\quad\quad\quad\quad p \quad r \\\mathrm { A } \begin{array} { l } a \\ { b }\end{array} \left[ \begin{array} { l l } 0&-2\\10&-7\end{array} \right] \\\end{array}$
E) $\begin{array} { l } \quad\quad\quad\quad\text { B } \\\quad\quad\quad\quad p \quad r \\\mathrm { A } \begin{array} { l } a \\ { b }\end{array} \left[ \begin{array} { l l } 0&-2 \\10&-7\end{array} \right] \\\end{array}$

Question 37

Two sectors of any country economy are (1) audio, video, and communication equipment and (2) electronic components and accessories. In 1998, the input-output table involving these two sectors was as follows (all figures are in millions of dollars): ​
Determine the production levels necessary in these two sectors to meet an external demand for $80,000 million of communication equipment and $110,000 million of electronic components. Round answers to two significant digits.
Equipment Sector production approximately __________ million, Components Sector production approximately __________ million.
​

Question 38

Four sectors of some economy are (1) Sector 1, (2) Sector 2, (3) Sector 3, and (4) Sector 4. The input-output table involving these four sectors was as follows (all figures are in millions of dollars): $$\begin{array} { | r | r | r | r | r | } \hline \text { To } & 1& 2 &3 & 4 \\\hline \text { From } \mathbf { 1 } & 174.3 & 30.7 & 120.4 & 14.1 \\\hline \mathbf { 2 } & 0 & 190.2 & 55.9 & 12.4 \\\hline \mathbf { 3 } & 2 & 40.9 & 1,418.1 & 1,242.1 \\\hline \mathbf { 4 } & 0.1 & 7.1 & 40.7 & 326.2 \\\hline \text { Total Output } & 3,278 & 2,188.5 & 6,541.1 & 4,065.7 \\\hline\end{array}$$
How much additional production by the Sector 1 is necessary to accommodate a $1,000 increase in the demand for the products of Sector 4 Round your answer to two decimal places.
$ __________ million

Question 39

Let $A = \left[ \begin{array} { l l } 0.9 & 0.2 \\0.4 & 0.5\end{array} \right]$ be the technology matrix, $X = \left[ \begin{array} { l } x _ { 1 } \\x _ { 2 }\end{array} \right]$ be the production vector, and $D = \left[ \begin{array} { c } 4,000 \\12,000\end{array} \right]$ be the external demand vector.

There would be 0.4 units of sector 2 needed to produce one unit of sector 2.

Question 40

Four sectors of some economy are (1) Sector 1, (2) Sector 2, (3) Sector 3, and (4) Sector 4. The input-output table involving these four sectors was as follows (all figures are in millions of dollars) ​
Determine how these four sectors would react to an increase in demand for Sector 1 production of $1,000 million. Round your answers to two decimal places.
Express the answer as a column matrix.
​

Question 41

Calculate the expected value of the game with the given payoff matrix using the mixed strategy supplied.
$P = \left[ \begin{array} { c c c c } 2 & 0 & - 1 & 2 \\- 1 & 0 & 0 & - 2 \\- 2 & 0 & 0 & 1 \\3 & 1 & - 1 & 1\end{array} \right]$ , $R = [ 1,0,0,0 ]$ , $C = [ 1,0,0,0 ] ^ { T }$

A) $e = 2$
B) $e = 1.5$
C) $e = 3.5$
D) $e = 3$
E) $e = 4$

Question 42

City Community College (CCC) plans to host Midtown Military Academy (MMA) for a wrestling tournament. Each school has three wrestlers in the 190 lb. weight class: CCC has Pablo, Sal, and Edison, while MMA has Carlos, Marcus and Noto. Pablo can beat Carlos and Marcus, Marcus can beat Sal and Edison, Sal can beat Carlos, Noto can beat Edison, while the other combinations will result in an even match. Set up a payoff matrix, and use reduction by dominance to decide which wrestler each team should choose as their champion. Does one school have an advantage over the other ?

A)Pablo vs. Noto; evenly matched
B) Sal vs. Marcus; Marcus wins
C) Sal vs. Noto; evenly matched
D) Pablo vs. Carlos; Pablo wins
E) Sal vs. Carlos; evenly matched

Question 43

Reduce the payoff matrix by dominance.
$\begin{array} { l } &B& \\\text { A } &\begin{array} { r } \mathrm { P } \\\mathrm { Q } \\\mathrm { R } \\\mathrm { P }\end{array} \left[ \begin{array} { c c c } a &b& c\\2&-2&-6\\5&0&3\\3&-3&10\\3&-6&-4\end{array} \right] \\\end{array}$

A) $\begin{array} { l } \quad\quad\quad\quad\text { B } \\\quad\quad\quad\quad a \quad b \\\mathrm { A } \begin{array} { l } q \\ { r }\end{array} \left[ \begin{array} { l l } 5&0\\3&-3\end{array} \right] \\\end{array}$
B) $\begin{array} { l } \quad\quad\quad\text { B } \\\quad\quad\quad b \quad \\\mathrm { A } \begin{array} { l } q\end{array} \left[ \begin{array} { l l } 0\end{array} \right] \\\end{array}$
C) $\begin{array} { l } \quad\quad\quad\text { B } \\\quad\quad\quad b \quad \\\mathrm { A } \begin{array} { l } q\end{array} \left[ \begin{array} { l l } -6\end{array} \right] \\\end{array}$
D) $\begin{array} { l } \quad\quad\quad\quad\text { B } \\\quad\quad\quad\quad a \quad c \\\mathrm { A } \begin{array} { l } q \\ { r }\end{array} \left[ \begin{array} { l l } -2&-6\\0&3\end{array} \right] \\\end{array}$
E) $\begin{array} { l } \quad\quad\quad\quad\text { B } \\\quad\quad\quad\quad a \quad c \\\mathrm { A } \begin{array} { l } q \\ { r }\end{array} \left[ \begin{array} { l l } 2&-2\\5&0\end{array} \right] \\\end{array}$

Question 44

A farmer has a choice of growing wheat, barley, or rice. Her success will depend on the weather, which could be dry, average, or wet. Her payoff matrix is as follows. $\begin{array}{|l|lrcr|}\hline & {\text { Weather }} \\\hline & & \text { Dry } & \text { Average } & \text { Wet } \\\text { Crop Choices } & \text { Wheat } & 20 & 20 & 10 \\& \text { Barley } & 10 & 15 & 20 \\& \text { Rice } & 10 & 20 & 20\\\hline\end{array}$ If the probability that the weather will be dry is 50%, the probability that it will be average is 10%, and the probability that it will be wet is 40%, what is the farmer's best choice of crop

A)Barley
B) Barley and Rice
C) Wheat
D) Wheat and Barley
E) Rice

Question 45

For the given row strategy R, find the optimal pure strategy (or strategies) the other player should use. Express the answer as a row or column matrix. $P = \left[ \begin{array} { c c c } 0 & - 1 & 5 \\2 & - 2 & 4 \\0 & 3 & 0 \\1 & 0 & - 5\end{array} \right]$ , $R = \left[ \frac { 4 } { 5 } , \frac { 1 } { 5 } , 0,0 \right]$

A)The strategy is $[ 0,1,0 ] ^ { T }$ ; the resulting expected value of the game is -1.2
B) The strategy is $[ 1,0,0 ] ^ { T }$ ; the resulting expected value of the game is -1.2
C) The strategy is $[ 0,1,0 ] ^ { T }$ ; the resulting expected value of the game is -0.7
D) The strategy is $[ 0,0,1 ] ^ { T }$ ; the resulting expected value of the game is -1.2
E) The strategy is $[ 0,1,0 ] ^ { T }$ ; the resulting expected value of the game is -0.2

Question 46

Reduce the payoff matrix by dominance.
​

Question 47

Your Abercrom B men's fashion outlet has a 60% chance of launching an expensive new line of used auto-mechanic dungarees (complete with grease stains) and a 40% chance of staying instead with its traditional torn military-style dungarees. Your rival across from you in the mall, Abercrom A, appears to be deciding between a line of torn gym shirts and a more daring line of "empty shirts" (that is, empty shirt boxes). Your corporate spies reveal that there is a 50% chance that Abercrom A will opt for the empty shirt option. The payoff matrix gives the number of customers your outlet can expect to gain from Abercrom A in each situation.
? $\begin{array} { l } \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\text { Abercrom A }\\\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \text { Torn } \quad \text { Empty } \\\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\text { Shirts } \quad\text { Shirts }\\\text { Abercrom B } \begin{array} { l } \text { Mechanics }\\\text { Military }\end{array} \left[ \begin{array} { l l } 10&-40\\-30&50\end{array} \right] \\\end{array}$ ?
How many customers can you expect to lose Round the answer to the nearest whole.
?
__________ customers

Question 48

In the Second World War, during the struggle for New Guinea, intelligence reports revealed that the Japanese were planning to move a troop and supply convoy from the port of Rabaul at the Eastern tip of New Britain to Lae, which lies just west of New Britain on New Guinea. It could either travel via a northern route which was plagued by poor visibility, or by a southern route, where the visibility was clear. General Kenney, who was the commander of the Allied Air Forces in the area, had the choice of concentrating reconnaissance aircraft on one route or the other, and bombing the Japanese convoy once it was sighted. Suppose that General Kenney had a third alternative: Splitting his reconnaissance aircraft between the two routes Kenney's staff drafted the following outcomes for his choices, where the payoffs are estimated days of bombing time: $$\begin{array}{l}\text { Japanese Commander's Strategies }\\\begin{array} { l | l | c | c | } \hline & & \text { NorthernRoute } & \text { SouthernRoute } \\\hline \text { Kenney's } & \text { Northern Route } & 2 & 2.5 \\\hline \text { Strategies } & \text { Split Reconnaissance } & 0.5 & 3 \\\hline & \text { Southern Route } & 1 & 3.5 \\\hline \end{array}\end{array}$$ What would you have recommended to General Kenney What would you have recommended to the Japanese Commander How much bombing time results if these recommendations are followed

A)General Kenney should use Split Reconnaissance and the Japanese Commander should use southern route; 3 days
B) General Kenney should use northern route and the Japanese Commander should use southern route; 2.5 days
C) Both commanders should use the northern route; 2 days
D) Both commanders should use the southern route; 1 days
E) General Kenney should use Split Reconnaissance and the Japanese Commander should use northern route; 0.5 days

Question 49

Reduce the payoff matrix by dominance.
​

Question 50

Your fast-food outlet, Burger Queen, has obtained a license to open branches in three closely situated South African cities: Brakpan, Nigel, and Springs. Your market surveys show that Brakpan and Nigel each provide a potential market of 2,500 burgers a day, while Springs provides a potential market of 1,000 burgers per day. Your company can only finance an outlet in one of those cities at the present time. Your main competitor, Burger Princess, has also obtained licenses for these cities, and is similarly planning to open only one outlet. If you both happen to locate at the same city, you will share the total business from all three cities equally, but if you locate in different cities, you will each get all the business in the cities in which you have located, plus half the business in the third city. The payoff is the number of burgers you will sell per day minus the number of burgers your competitor will sell per day. Set up the payoff matrix.

A) $\begin{array} { l } \quad\quad\quad\quad\quad\quad\text { Your Competitor} \\\quad\quad\quad\quad\quad B\quad\quad\quad N\quad\quad\quad\quad S \\\text { you } \begin{array} { l } B \\ { N }\\S\end{array} \left[ \begin{array} { l l } 0&0&-1,500\\0&0&-1,500\\1,500&1,500&0\end{array} \right] \\\end{array}$
B) $\begin{array} { l } \quad\quad\quad\quad\quad\quad\text { Your Competitor} \\\quad\quad\quad\quad\quad B\quad\quad\quad N\quad\quad\quad\quad S \\\text { you } \begin{array} { l } B \\ { N }\\S\end{array} \left[ \begin{array} { l l } 0&0&1,500\\0&0&1,500\\-1,500&-1,500&0\end{array} \right] \\\end{array}$

C) $\begin{array} { l } \quad\quad\quad\quad\quad\quad\text { Your Competitor} \\\quad\quad\quad\quad\quad B\quad\quad\quad N\quad\quad\quad\quad S \\\text { you } \begin{array} { l } B \\ { N }\\S\end{array} \left[ \begin{array} { l l } -1,500&0&0\\0&-1,500&0\\0&0&-1,500\end{array} \right] \\\end{array}$

D) $\begin{array} { l } \quad\quad\quad\quad\quad\quad\text { Your Competitor} \\\quad\quad\quad\quad\quad B\quad\quad\quad N\quad\quad\quad\quad S \\\text { you } \begin{array} { l } B \\ { N }\\S\end{array} \left[ \begin{array} { l l } 0&1,500&1,500\\-1,500&0&1,500\\-1,500&-1,500&0\end{array} \right] \\\end{array}$

E) $\begin{array} { l } \quad\quad\quad\quad\quad\quad\text { Your Competitor} \\\quad\quad\quad\quad\quad B\quad\quad\quad N\quad\quad\quad\quad S \\\text { you } \begin{array} { l } B \\ { N }\\S\end{array} \left[ \begin{array} { l l } 1,500&0&0\\0&1,500&0\\0&0&1,500\end{array} \right] \\\end{array}$

Question 51

Decide whether the game is strictly determined. If it is, give the players' optimal pure strategies and the value of the game.
$\begin{array} { l } &B& \\\text { A } &\begin{array} { r } \mathrm { P } \\\mathrm { Q } \\\mathrm { R } \\\mathrm { P }\end{array} \left[ \begin{array} { c c c } a &b& c\\-3&-5&10\\1&1&0\\-1&-2&-3\\1&1&-1\end{array} \right] \\\end{array}$

A)A's optimal strategy is Q; B's optimal strategy is a; value: 0.5
B) A's optimal strategy is P; B's optimal strategy is a; value: 1
C) A's optimal strategy is P; B's optimal strategy is b; value: 1
D) A's optimal strategy is R; B's optimal strategy is c; value: 3
E) The game is not strictly determined

Question 52

You and your friend have come up with the following simple game to pass the time: at each round, you simultaneously call "heads" or "tails". If you have both called the same thing, your friend wins one point; if your calls differ, you win one point. Set up the payoff matrix.

A) $\begin{array} { l } \quad\quad\quad\quad\text { friend } \\\quad\quad\quad\quad\quad H \quad T \\\mathrm { you} \begin{array} { l } H \\ { T }\end{array} \left[ \begin{array} { l l } -1&1\\1&-1\end{array} \right] \\\end{array}$
B) $\begin{array} { l } \quad\quad\quad\quad\text { friend } \\\quad\quad\quad\quad\quad H \quad T \\\mathrm { you} \begin{array} { l } H \\ { T }\end{array} \left[ \begin{array} { l l } 1&1\\-1&-1\end{array} \right] \\\end{array}$

C) $\begin{array} { l } \quad\quad\quad\quad\text { friend } \\\quad\quad\quad\quad\quad H \quad T \\\mathrm { you} \begin{array} { l } H \\ { T }\end{array} \left[ \begin{array} { l l } -1&-1\\1&1\end{array} \right] \\\end{array}$

D) $\begin{array} { l } \quad\quad\quad\quad\text { friend } \\\quad\quad\quad\quad\quad H \quad T \\\mathrm { you} \begin{array} { l } H \\ { T }\end{array} \left[ \begin{array} { l l } 1&-1\\-1&1\end{array} \right] \\\end{array}$

E) $\begin{array} { l } \quad\quad\quad\quad\text { friend } \\\quad\quad\quad\quad\quad H \quad T \\\mathrm { you} \begin{array} { l } H \\ { T }\end{array} \left[ \begin{array} { l l } 2&-1\\-1&2\end{array} \right] \\\end{array}$

Question 53

Decide whether the game is strictly determined. If it is, give the players' optimal pure strategies and the value of the game.
$\begin{array} { l } \quad\quad\quad\quad\text { B } \\\quad\quad\quad\quad p \quad q\quad r \\\mathrm { A } \begin{array} { l } a \\ { b }\end{array} \left[ \begin{array} { l l } -3&2&-4\\-3&3&-3\end{array} \right] \\\end{array}$

A)A's optimal strategy is a; B's optimal strategy is q; value: 1
B) A's optimal strategy is b; B's optimal strategy is q; value: 1
C) A's optimal strategy is b; B's optimal strategy is p; value: 1
D) A's optimal strategy is a; B's optimal strategy is p; value: 1.5
E) The game is not strictly determined

Question 54

Your Abercrom B men's fashion outlet has a 40% chance of launching an expensive new line of used auto-mechanic dungarees (complete with grease stains) and a 60% chance of staying instead with its traditional torn military-style dungarees. Your rival across from you in the mall, Abercrom A, appears to be deciding between a line of torn gym shirts and a more daring line of "empty shirts" (that is, empty shirt boxes). Your corporate spies reveal that there is a 40% chance that Abercrom A will opt for the empty shirt option. The payoff matrix gives the number of customers your outlet can expect to gain from Abercrom A in each situation.
What is the expected resulting effect on your customer base Round the answer to the nearest whole number.

A)You can expect to lose 3 customers.
B) You can expect to lose 4 customers.
C) You can expect to lose 5 customers.
D) You can expect to gain 3 customers.
E) You can expect to gain 4 customers.

Question 55

Find the optimal mixed row strategy, the optimal mixed column strategy, and the expected value of the game in the event that each player uses his or her optimal mixed strategy.
$P = \left[ \begin{array} { c c } - 7 & - 8 \\- 8 & 7\end{array} \right]$

A) $R = \left[ \frac { 15 } { 16 } , \frac { 1 } { 16 } \right]$ , $C = \left[ \frac { 15 } { 16 } , \frac { 1 } { 16 } \right] ^ { T }$ , $e = - \frac { 15 } { 16 }$
B) $R = \left[ \frac { 15 } { 16 } , \frac { 1 } { 16 } \right]$ , $C = \left[ \frac { 15 } { 16 } , \frac { 1 } { 16 } \right] ^ { T }$ , $e = - \frac { 113 } { 16 }$
C) $R = \left[ \frac { 1 } { 16 } , \frac { 15 } { 16 } \right]$ , $C = \left[ \frac { 15 } { 16 } , \frac { 1 } { 16 } \right] ^ { T }$ , $e = - \frac { 113 } { 16 }$
D) $R = \left[ \frac { 15 } { 16 } , \frac { 1 } { 16 } \right]$ , $C = \left[ \frac { 1 } { 16 } , \frac { 15 } { 16 } \right] ^ { T }$ , $e = - \frac { 113 } { 16 }$
E) $R = \left[ \frac { 1 } { 16 } , \frac { 15 } { 16 } \right]$ , $C = \left[ \frac { 15 } { 16 } , \frac { 1 } { 16 } \right] ^ { T }$ , $e = - \frac { 15 } { 16 }$

Question 56

Decide whether the game is strictly determined. If it is, give the players' optimal pure strategies and the value of the game.
$\begin{array} { l } \quad\quad\quad\quad\text { B } \\\quad\quad\quad\quad p \quad q \\\mathrm { A } \begin{array} { l } a \\{ b }\end{array} \left[ \begin{array} { l l } 1&1 \\3&-5\end{array} \right] \\\end{array}$

A)A's optimal strategy is b; B's optimal strategy is q; value: 0.5
B) A's optimal strategy is a; B's optimal strategy is p; value: 2
C) A's optimal strategy is a; B's optimal strategy is q; value: 1
D) A's optimal strategy is b; B's optimal strategy is p; value: 1
E) The game is not strictly determined.

Question 57

A manufacturer of electrical machinery is located in a cramped, though low-rent, factory close to the center of a large city. The firm needs to expand, and it could do so in one of three ways: (1) remain where it is and install new equipment, (2) move to a suburban site in the same city, or (3) relocate in a different part of the country where labor is cheaper. Its decision will be influenced by the fact that one of the following will happen: (I) the government may introduce a program of equipment grants, (II) a new suburban highway may be built, or (III) the government may institute a policy of financial help to companies who move into regions of high unemployment. The value to the company of each combination is given in the following payoff matrix. $$\begin{array} { | l | l | l | l | } \hline & \text { I } & \text { II } & \text { III } \\\hline \mathbf { 1 } & 200 & 150 & 140 \\\hline \mathbf { 2 } & 130 & 220 & 130 \\\hline \mathbf { 3 } & 110 & 110 & 220 \\\hline\end{array}$$ If the manufacturer judges that there is a 60% probability that the government will go with option I, a 30% probability that they will go with option II, and a 10% probability that they will go with option III, what is the manufacturer's best option

A)Option 1
B) Option 3
C) Option 1 and Option 2
D) Option 2
E) Option 1 and Option 3

Question 58

For the given row strategy R, find the optimal pure strategy (or strategies) the other player should use. Express the answer as a row or column matrix. Also determine the resulting expected value of the game. $$P = \left[ \begin{array} { c c c } 0 & - 1 & 5 \\2 & - 2 & 4 \\0 & 3 & 0 \\1 & 0 & - 5\end{array} \right]$$ , $R = [ 0.7,0.3,0,0 ]$

A)The strategy is $[ 0,1,0 ] ^ { T }$ ; the resulting expected value of the game is -0.3
B) The strategy is $[ 1,0,0 ] ^ { T }$ ; the resulting expected value of the game is -1.3
C) The strategy is $[ 0,1,0 ] ^ { T }$ ; the resulting expected value of the game is -1.3
D) The strategy is $[ 0,0,1 ] ^ { T }$ ; the resulting expected value of the game is -1.3
E) The strategy is $[ 0,1,0 ] ^ { T }$ ; the resulting expected value of the game is -0.8

Question 59

Find the optimal mixed row strategy, the optimal mixed column strategy, and the expected value of the game in the event that each player uses his or her optimal mixed strategy.
$P = \left[ \begin{array} { l l } - 7 & - 2 \\- 2 & - 3\end{array} \right]$

A) $R = \left[ \frac { 1 } { 6 } , - \frac { 5 } { 6 } \right]$ , $C = \left[ \frac { 1 } { 6 } , - \frac { 5 } { 6 } \right] ^ { T }$ , $e = - \frac { 5 } { 6 }$
B) $R = \left[ \frac { 1 } { 6 } , - \frac { 5 } { 6 } \right]$ , $C = \left[ \frac { 5 } { 6 } , - \frac { 1 } { 6 } \right] ^ { T }$ , $e = - \frac { 17 } { 6 }$
C) $R = \left[ \frac { 5 } { 6 } , - \frac { 1 } { 6 } \right]$ , $C = \left[ \frac { 1 } { 6 } , - \frac { 5 } { 6 } \right] ^ { T }$ , $e = - \frac { 17 } { 6 }$
D) $R = \left[ \frac { 1 } { 6 } , \frac { 5 } { 6 } \right]$ , $C = \left[ \frac { 1 } { 6 } , \frac { 5 } { 6 } \right] ^ { T }$ , $e = - \frac { 17 } { 6 }$
E) $R = \left[ \frac { 5 } { 6 } , - \frac { 1 } { 6 } \right]$ , $C = \left[ \frac { 5 } { 6 } , - \frac { 1 } { 6 } \right] ^ { T }$ , $e = - \frac { 5 } { 6 }$

Question 60

You are deciding whether to invade England, Sweden or Norway, and your opponent is simultaneously deciding which of these three countries to defend. If you invade a country that your opponent is defending, you will be defeated (payoff: -1), but if you invade a country your opponent is not defending, you will be successful (payoff: +1). Set up the payoff matrix.

A) $\begin{array} { l } \quad\quad\text { Your Opponent Defends } \\\quad\quad\quad\quad\quad\quad\quad\quad\quad E\quad S\quad N \\\text { you Invade } \begin{array} { l } E \\ { S }\\N\end{array} \left[ \begin{array} { l l } 1&1&-1\\1&-1&1\\-1&1&1\end{array} \right] \\\end{array}$
B) $\begin{array} { l } \quad\quad\text { Your Opponent Defends } \\\quad\quad\quad\quad\quad\quad\quad\quad\quad E\quad S\quad N \\\text { you Invade } \begin{array} { l } E \\ { S }\\N\end{array} \left[ \begin{array} { l l } -1&1&1\\1&-1&1\\1&1&-1\end{array} \right] \\\end{array}$

C) $\begin{array} { l } \quad\quad\text { Your Opponent Defends } \\\quad\quad\quad\quad\quad\quad\quad\quad\quad E\quad S\quad N \\\text { you Invade } \begin{array} { l } E \\ { S }\\N\end{array} \left[ \begin{array} { l l } 1&-1&1\\-1&1&1\\1&1&-1\end{array} \right] \\\end{array}$

D) $\begin{array} { l } \quad\quad\text { Your Opponent Defends } \\\quad\quad\quad\quad\quad\quad\quad\quad\quad E\quad S\quad N \\\text { you Invade } \begin{array} { l } E \\ { S }\\N\end{array} \left[ \begin{array} { l l } 1&1&-1\\-1&1&1\\1&-1&1\end{array} \right] \\\end{array}$

E) $\begin{array} { l } \quad\quad\text { Your Opponent Defends } \\\quad\quad\quad\quad\quad\quad\quad\quad\quad E\quad S\quad N \\\text { you Invade } \begin{array} { l } E \\ { S }\\N\end{array} \left[ \begin{array} { l l } 1&-1&-1\\-1&1&-1\\-1&-1&1\end{array} \right] \\\end{array}$

Question 61

Calculate the expected value of the game with the given payoff matrix using the mixed strategy supplied.
​
​ , ,

Question 62

In the Second World War, during the struggle for New Guinea, intelligence reports revealed that the Japanese were planning to move a troop and supply convoy from the port of Rabaul at the Eastern tip of New Britain to Lae, which lies just west of New Britain on New Guinea. It could either travel via a northern route which was plagued by poor visibility, or by a southern route, where the visibility was clear. General Kenney, who was the commander of the Allied Air Forces in the area, had the choice of concentrating reconnaissance aircraft on one route or the other, and bombing the Japanese convoy once it was sighted. Suppose that General Kenney had a third alternative: Splitting his reconnaissance aircraft between the two routes Kenney's staff drafted the following outcomes for his choices, where the payoffs are estimated days of bombing time: $\begin{array} { l }&\text { Japanese Commander's Strategies }\\\begin{array} { l } \text { Kenney's}\\ \text {Strategies }\\\end{array}&\begin{array}{|l|c|c|}\hline & \text { NorthernRoute } & \text { SouthernRoute } \\\hline \text { Northern Route } & 2 & 2.5 \\\hline \text { Split Reconnaissance } & 0.5 & 2 \\\hline \text { Southern Route } & 1 & 4 \\\hline\end{array}\end{array}$
What would you have recommended to General Kenney

__________ (Northern Route or Split Reconnaissance or Southern Route)
What would you have recommended to the Japanese Commander

__________ (Northern Route or Southern Route)
How much bombing time results if these recommendations are followed

__________days

Question 63

When you bet on a racehorse with odds of m-n, you stand to win m dollars for every bet of n dollars if your horse wins; for instance, if the horse you bet is running at 5-2 and wins, you will win $5 for every $2 you bet. (Thus a $2 bet will return $7.). Here are some actual odds from a 1992 race at Belmont Park, NY. The favorite at 4-1 was Pleasant Tap. The second choice was Thunder Rumble at 7-2, while the third choice was Strike the Gold at 7-2. Assume you are making a $20 bet on one of these horses. The payoffs are your winnings. (If your horse does not win, you lose your entire bet. Of course, it is possible for none of your horses to win.) Suppose that just before the race, there has been frantic betting on Thunder Rumble, with the result that the odds have dropped to 1-5. The odds on the other two horses remain unchanged. Set up the payoff matrix with your bet as the row player and winner as the column player.
​
Let the first column correspond to Pleasant Tap, the second one to Thunder Rumble, the third one to Strike the Gold and the first row correspond to Pleasant Tap, the second one to Thunder Rumble, the third one to Strike the Gold .

Question 64

Which of the following is a true statement

A)If A and C are matrices and $A C = 1$ , then $C A = I$ .
B) Given $M = \left[ \begin{array} { l l } h & c \\g & b\end{array} \right]$ , if $h b - c g = 0$ , then M has an inverse.
C) If A, X, and C are matrices, and $A X = C$ , then $X = C A ^ { - 1 }$ .
D) If A and C are matrices and $A C = 1$ , then $C A = I$ and if A, X, and C are matrices, and $A X = C$ , then $X = C A ^ { - 1 }$ is true.
E) None of these statements is true.

Question 65

A farmer has a choice of growing wheat, barley, or rice. Her success will depend on the weather, which could be dry, average, or wet. Her payoff matrix is as follows. $\begin{array}{l}&\text { Weather }\\\begin{array} { l } \text {Crop }\\ \text { Choices}\\\end{array}&\begin{array}{|l|c|c|c|}\hline & \text { Dry } & \text { Average } & \text { Wet } \\\hline \text { Wheat } & 20 & 15 & 10 \\\hline \text { Barley } & 20 & 15 & 20 \\\hline \text { Rice } & 20 & 15 & 15 \\\hline\end{array}\end{array}$
If the probability that the weather will be dry is 20%, the probability that it will be average is 60%, and the probability that it will be wet is 20%, what is the farmer's best choice of crop

Question 66

Use matrix inversion to solve the given system of linear equations.
$\begin{array} { r } 2 x + 5 y = 5 \\5 x + 13 y = 2\end{array}$

A) $x = 55$ , $y = - 21$
B) $x = 56$ , $y = - 21$
C) $x = 56$ , $y = - 18$
D) $x = 55$ , $y = - 18$
E) The system is inconsistent.

Question 67

Use matrix inversion to solve the system of linear equations. $\begin{aligned}x + 2 y + z & = 8 \\x + y + z & = 7 \\3 x + y + z & = 25\end{aligned}$

A) $x = 7$ , $y = 4$ , $z = - 1$
B) $x = 12$ , $y = 3$ , $z = - 5$
C) $x = 11$ , $y = - 1$ , $z = 0$
D) $x = 9$ , $y = 1$ , $z = - 3$
E) The system is inconsistent.

Question 68

A manufacturer of electrical machinery is located in a cramped, though low-rent, factory close to the center of a large city. The firm needs to expand, and it could do so in one of three ways: (1) remain where it is and install new equipment, (2) move to a suburban site in the same city, or (3) relocate in a different part of the country where labor is cheaper. Its decision will be influenced by the fact that one of the following will happen: (I) the government may introduce a program of equipment grants, (II) a new suburban highway may be built, or (III) the government may institute a policy of financial help to companies who move into regions of high unemployment. The value to the company of each combination is given in the following payoff matrix. $$\begin{array} { | l | l | l | l | } \hline & \text { I } & \text { II } & \text { III } \\\hline \mathbf { 1 } & 170 & 120 & 130 \\\hline \mathbf { 2 } & 130 & 220 & 140 \\\hline \mathbf { 3 } & 130 & 120 & 220 \\\hline\end{array}$$
If the manufacturer judges that there is a 10% probability that the government will go with option I, a 10% probability that they will go with option II, and a 80% probability that they will go with option III, what is the manufacturer's best option

Question 69

Use row reduction to find the inverse of the given matrix, if it exists. Check your answers by multiplication.
$\left[ \begin{array} { l l } 3 & 1 \\5 & 2\end{array} \right]$

A) $\left[ \begin{array} { c c } 2 & - 1 \\- 5 & 3\end{array} \right]$
B) $\left[ \begin{array} { c c } - 3 & 1 \\5 & - 2\end{array} \right]$
C) $\left[ \begin{array} { c c } 2 & 5 \\- 1 & 3\end{array} \right]$
D) $\left[ \begin{array} { c c } 1 & - 3 \\- 2 & 5\end{array} \right]$
E) The inverse matrix does not exist.

Question 70

Use the row reduction method to find the inverse, if it exists.
$A = \left[ \begin{array} { l l } 2 & 5 \\1 & 3\end{array} \right]$

A) $\left[ \begin{array} { l l } - 2 & - 5 \\- 1 & - 3\end{array} \right]$
B) $\left[ \begin{array} { c c } - 3 & 5 \\1 & - 2\end{array} \right]$
C) $\left[ \begin{array} { l l } \frac { 1 } { 2 } & 0 \\0 & \frac { 1 } { 3 }\end{array} \right]$
D) $\left[ \begin{array} { c c } 3 & - 5 \\- 1 & 2\end{array} \right]$
E) no inverse exists

Question 71

Decide whether the game is strictly determined. $\begin{array} { l } \quad\quad\quad\quad\text { B } \\\quad\quad\quad\quad p \quad q \quad r \\\mathrm { A } \begin{array} { l } a \\ { b }\end{array} \left[ \begin{array} { l l } -2&2&-5\\-2&4&-2\end{array} \right] \\\end{array}$ __________ (answer strictly determined or not strictly determined)
If it is, what are the players' optimal pure strategies
A __________
B __________

What is the value of the game __________

Question 72

Determine which pair of matrices is an inverse pair.

A) $\left[ \begin{array} { l l } 0 & 3 \\3 & 0\end{array} \right] \left[ \begin{array} { c c } 0 & \frac { 1 } { 3 } \\\frac { 1 } { 3 } & 0\end{array} \right]$
B) $\left[ \begin{array} { l l } 3 & 3 \\0 & 3\end{array} \right] \left[ \begin{array} { c c } 0 & - 3 \\- 3 & 3\end{array} \right]$
C) $\left[ \begin{array} { l l } 0 & 3 \\3 & 0\end{array} \right] \left[ \begin{array} { l l } 0 & 3 \\3 & 0\end{array} \right]$
D) $\left[ \begin{array} { l l } 3 & 0 \\0 & 3\end{array} \right] \left[ \begin{array} { c c } 0 & \frac { 1 } { 3 } \\\frac { 1 } { 3 } & 0\end{array} \right]$
E) None of these

Question 73

Compute the determinant of the given matrix.
$\left[ \begin{array} { l l } 3 & - 1 \\5 & - 5\end{array} \right]$

A)26
B) -10
C) 25
D) -1
E) 1

Question 74

Your fast-food outlet, Burger Queen, has obtained a license to open branches in three closely situated South African cities: Brakpan, Nigel, and Springs. Your market surveys show that Brakpan and Nigel each provide a potential market of 2,500 burgers a day, while Springs provides a potential market of 1,000 burgers per day. Your company can only finance an outlet in one of those cities at the present time. Your main competitor, Burger Princess, has also obtained licenses for these cities, and is similarly planning to open only one outlet. If you both happen to locate at the same city, you will share the total business from all three cities equally, but if you locate in different cities, you will each get all the business in the cities in which you have located, plus half the business in the third city. The payoff is the number of burgers you will sell per day minus the number of burgers your competitor will sell per day. Set up the payoff matrix with you as the row player and your opponent as the column player.
​
Let the first column correspond to Brakpan, the second one to Nigel, the third one to Springs and the first row correspond to the Brakpan, the second one to the Nigel, the third one to Springs.

Question 75

Use row reduction to find the inverse of the given matrix, if it exists. Check your answer by multiplication.
$\left[ \begin{array} { c c c } 1 & 9 & 10 \\0 & 9 & 10 \\1 & 0 & 1\end{array} \right]$

A) $\left[ \begin{array} { c c c } - \frac { 10 } { 9 } & - 1 & \frac { 10 } { 9 } \\- 1 & 1 & - 1\end{array} \right]$
B) $\left[ \begin{array} { c c c } \frac { 10 } { 9 } & - 1 & - \frac { 10 } { 9 } \\- 1 & 1 & 1\end{array} \right]$
C) $\left[ \begin{array} { c c c } - \frac { 10 } { 9 } & - 1 & \frac { 10 } { 9 } \\1 & 1 & - 1\end{array} \right]$
D) $\left[ \begin{array} { c c c } 1 & 9 & - 10 \\0 & 9 & - 10 \\1 & 0 & 1\end{array} \right]$
E) The inverse matrix does not exist.

Question 76

Decide whether the game is strictly determined.

$\begin{array} { l } \quad\quad\quad\quad\text { B } \\\quad\quad\quad\quad p \quad q \\\mathrm { A } \begin{array} { l } a \\ { b }\end{array} \left[ \begin{array} { l l } 1&1\\4&-3\end{array} \right] \\\end{array}$

__________ (strictly determined or not strictly determined)
If it is, what are the players' optimal pure strategies
A __________
B __________

What is the value of the game __________

Question 77

You and your friend have come up with the following simple game to pass the time: at each round, you simultaneously call "heads" or "tails". If you have both called the same thing, your friend wins one point; if your calls differ, you win one point. Set up the payoff matrix with you as the row player and your friend as the column player.

Question 78

Decide whether the game is strictly determined. $\begin{array} { l } &B& \\\text { A } &\begin{array} { r } \mathrm { P } \\\mathrm { Q } \\\mathrm { R } \\\mathrm { P }\end{array} \left[ \begin{array} { c c c } a &b& c\\- 2 & - 5 & 9 \\1 & 1 & 0 \\- 1 & - 2 & - 3 \\1 & 1 & - 1\end{array} \right] \\\end{array}$ __________ (answer strictly determined or not strictly determined)
If it is, what are the players' optimal pure strategies
A __________
B __________

What is the value of the game __________

Question 79

You are deciding whether to invade Japan, Sweden or Canada, and your opponent is simultaneously deciding which of these three countries to defend. If you invade a country that your opponent is defending, you will be defeated (payoff: -2), but if you invade a country your opponent is not defending, you will be successful (payoff: +2). Set up the payoff matrix with you as the row player and your opponent as the column player.

Question 80

Find the inverse of the given matrix, if it exists.
$\left[ \begin{array} { c c c } 1 & 1 & 1 \\1 & 0 & 5 \\1 & - 1 & 1\end{array} \right]$

A) $\left[ \begin{array} { c c c } \frac { 5 } { 8 } & \frac { 1 } { 8 } & \frac { 5 } { 8 } \\\frac { 1 } { 2 } & 8 & - \frac { 1 } { 2 } \\\frac { 1 } { 8 } & \frac { 1 } { 4 } & - \frac { 1 } { 8 }\end{array} \right]$
B) $\left[ \begin{array} { c c c } - \frac { 5 } { 8 } & - \frac { 1 } { 4 } & \frac { 5 } { 8 } \\- \frac { 1 } { 2 } & 0 & - \frac { 1 } { 2 } \\- \frac { 1 } { 8 } & \frac { 1 } { 4 } & - \frac { 1 } { 8 }\end{array} \right]$
C) $\left[ \begin{array} { c c c } - \frac { 5 } { 8 } & - \frac { 1 } { 4 } & \frac { 5 } { 8 } \\- \frac { 1 } { 2 } & 0 & - \frac { 1 } { 2 } \\- \frac { 1 } { 8 } & \frac { 1 } { 4 } & 8\end{array} \right]$
D) $\left[ \begin{array} { c c c } \frac { 5 } { 8 } & - \frac { 1 } { 4 } & \frac { 5 } { 8 } \\\frac { 1 } { 2 } & 0 & - \frac { 1 } { 2 } \\- \frac { 1 } { 8 } & \frac { 1 } { 4 } & - \frac { 1 } { 8 }\end{array} \right]$
E) The inverse matrix does not exist.