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 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

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. 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.

Reduce the payoff matrix by dominance. \[\begin{array} { l } B\\ \mathrm { A } \begin{array} { c } p \\ r \\ s \\ s \end{array} \left[ \begin{array} { c c c } a&b&c\\ 1 & - 2 & - 5 \\ 4 & 0 & 2 \\ 3 & - 3 & 10 \\ 3 & - 5 & - 4 \end{array} \right] \\ \end{array}\]

The answer of Reduce the payoff matrix by dominance. \[\begin{array}...

Use matrix inversion to solve the given system of linear equations. $\begin{array} { r } 4 x + 15 y = 1 \\ x + 4 y = 1 \end{array}$ $x=\frac{\text { }}{\text { }}$ $y=\frac{\text { }}{\text { }}$

The answer of Use matrix inversion to solve the given...

Obtain the technology matrix from the following input-output table. \[\begin{array} { | r | c | c | c | } \hline \text { to } & \mathbf { A } & \mathbf { B } & \mathbf { C } \\ \hline \text { from } \mathbf { A } & 0 & 200 & 2,100 \\ \hline \mathbf { B } & 500 & 300 & 2,100 \\ \hline \mathbf { C } & 0 & 0 & 4,200 \\ \hline \text { Total Output } & 800 & 2,000 & 3,000 \\ \hline \end{array}\]

The answer of Obtain the technology matrix from the following...

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

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

Exam 5: Matrix Algebra and Applications

Translate the given system of equations into matrix form. $\left\{ \begin{array} { r } x + y - z = 1 \\8 x + y + z = 3 \\\frac { 3 x } { 4 } + \frac { z } { 2 } = 4\end{array} \right\}$

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Question 1

Correct Answer:

Verified

E

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. To Choral Society Football Club From Choral Society 20 35 Football Club 35 30 Total Output 100 200 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

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Question 2

Correct Answer:

Verified

A

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

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Question 3

Correct Answer:

Verified

D

Use row reduction to find the inverse of the given matrix, if it exists, and check your answer by multiplication. $\left[ \begin{array} { l l } 2 & 2 \\8 & 8\end{array} \right]$

(Multiple Choice)

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Question 4

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

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Question 5

Decide whether the game is strictly determined. If it is, give the players' optimal pure strategies and the value of the game. B A a b c -3 -5 10 1 1 0 -1 -2 -3 1 1 -1

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Question 6

Use matrix inversion to solve the given system of linear equations. 2x+5y=5 5x+13y=2

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Question 7

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

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Question 8

Translate the matrix equations into a system of linear equations. $\left[ \begin{array} { c c c } 1 & - 1 & 5 \\- \frac { 1 } { 3 } & - 3 & \frac { 1 } { 3 } \\2 & 0 & 1\end{array} \right] \cdot \left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } - 5 \\- 9 \\7\end{array} \right]$

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Question 9

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

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Question 10

Reduce the payoff matrix by dominance. B p r s s a b c 1 -2 -5 4 0 2 3 -3 10 3 -5 -4

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Question 11

Microbucks Computers makes two computers, the Pomegranate II and the Pomegranate Classic, at two different factories. The Pom II requires 2 processor chips, 18 memory chips, and 20 vacuum tubes, and the Pom Classic requires 2 processor chips, 6 memory chips, and 50 vacuum tubes. At the beginning of the year, Microbucks has in stock 800 processor chips, 6,000 memory chips, and 13,000 vacuum tubes at the Pom II factory and 400 processor chips, 4,000 memory chips, and 40,000 vacuum tubes at the Pom Classic factory. It manufactures 50 Pom IIs and 50 Pom Classics each month. Find the company's inventory of parts after 2 months, using matrix operations.

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Question 12

Use row reduction to find the inverse of the given matrix, if it exists. Check your answer by multiplication. $\left[ \begin{array} { l l l } 1 & 2 & 3 \\0 & 2 & 3 \\1 & 0 & 1\end{array} \right]$

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Question 13

Use matrix inversion to solve the given system of linear equations. 4x+15y=1 x+4y=1 $x=\frac{\text { }}{\text { }}$ $y=\frac{\text { }}{\text { }}$

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Question 14

Find the matrix product, if possible. $\left[ \begin{array} { c c c } 3 & - 5 & 0 \\2 & 4 & 1\end{array} \right] \cdot \left[ \begin{array} { l l } 4 & 5 \\2 & 0\end{array} \right]$

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Question 15

Compute the determinant of the matrix. If the determinant is nonzero, use the formula for inverting a $2 \times 2$ matrix to calculate the inverse of the given matrix. $\left[ \begin{array} { c c } 1 & 2 \\2 & - 1\end{array} \right]$

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Question 16

Obtain the technology matrix from the following input-output table. to from 0 200 2,100 500 300 2,100 0 0 4,200 Total Output 800 2,000 3,000

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Question 17

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. Weather Dry Average Wet Crop Choices Wheat 20 20 10 Barley 10 15 20 Rice 10 20 20 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

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Question 18

If $A = \left[ \begin{array} { l l } 3 & 7 \\2 & 4\end{array} \right]$ and $B = \left[ \begin{array} { c c } - 3 & 6 \\4 & - 2\end{array} \right]$ , then find $A - 2 B$ .

(Multiple Choice)

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Question 19

Reduce the payoff matrix by dominance. B A a b c 2 -2 -6 5 0 3 3 -3 10 3 -6 -4

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Question 20

Translate the given system of equations into matrix form. $\left\{ \begin{array} { r } x + y - z = 1 \\8 x + y + z = 3 \\\frac { 3 x } { 4 } + \frac { z } { 2 } = 4\end{array} \right\}$

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

Use row reduction to find the inverse of the given matrix, if it exists, and check your answer by multiplication. $\left[ \begin{array} { l l } 2 & 2 \\8 & 8\end{array} \right]$

Decide whether the game is strictly determined. If it is, give the players' optimal pure strategies and the value of the game. B A a b c -3 -5 10 1 1 0 -1 -2 -3 1 1 -1

Use matrix inversion to solve the given system of linear equations. 2x+5y=5 5x+13y=2

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

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

Reduce the payoff matrix by dominance. B p r s s a b c 1 -2 -5 4 0 2 3 -3 10 3 -5 -4

Use row reduction to find the inverse of the given matrix, if it exists. Check your answer by multiplication. $\left[ \begin{array} { l l l } 1 & 2 & 3 \\0 & 2 & 3 \\1 & 0 & 1\end{array} \right]$

Use matrix inversion to solve the given system of linear equations. 4x+15y=1 x+4y=1 $x=\frac{\text { }}{\text { }}$ $y=\frac{\text { }}{\text { }}$

Find the matrix product, if possible. $\left[ \begin{array} { c c c } 3 & - 5 & 0 \\2 & 4 & 1\end{array} \right] \cdot \left[ \begin{array} { l l } 4 & 5 \\2 & 0\end{array} \right]$

Compute the determinant of the matrix. If the determinant is nonzero, use the formula for inverting a $2 \times 2$ matrix to calculate the inverse of the given matrix. $\left[ \begin{array} { c c } 1 & 2 \\2 & - 1\end{array} \right]$

Obtain the technology matrix from the following input-output table. to from 0 200 2,100 500 300 2,100 0 0 4,200 Total Output 800 2,000 3,000

If $A = \left[ \begin{array} { l l } 3 & 7 \\2 & 4\end{array} \right]$ and $B = \left[ \begin{array} { c c } - 3 & 6 \\4 & - 2\end{array} \right]$ , then find $A - 2 B$ .

Reduce the payoff matrix by dominance. B A a b c 2 -2 -6 5 0 3 3 -3 10 3 -6 -4

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Exam 5: Matrix Algebra and Applications

Translate the given system of equations into matrix form. {x+y−z=18x+y+z=33x4+z2=4}\left\{ \begin{array} { r } x + y - z = 1 \\8 x + y + z = 3 \\\frac { 3 x } { 4 } + \frac { z } { 2 } = 4\end{array} \right\}⎩⎨⎧​x+y−z=18x+y+z=343x​+2z​=4​⎭⎬⎫​

Use the row reduction method to find the inverse, if it exists. A=[2513]A = \left[ \begin{array} { l l } 2 & 5 \\1 & 3\end{array} \right]A=[21​53​]

Use row reduction to find the inverse of the given matrix, if it exists, and check your answer by multiplication. [2288]\left[ \begin{array} { l l } 2 & 2 \\8 & 8\end{array} \right][28​28​]

Decide whether the game is strictly determined. If it is, give the players' optimal pure strategies and the value of the game. B A a b c -3 -5 10 1 1 0 -1 -2 -3 1 1 -1

Use matrix inversion to solve the given system of linear equations. 2x+5y=5 5x+13y=2

Reduce the payoff matrix by dominance. ​ B p r s s a b c 1 -2 -5 4 0 2 3 -3 10 3 -5 -4

Use row reduction to find the inverse of the given matrix, if it exists. Check your answer by multiplication. ​ [123023101]\left[ \begin{array} { l l l } 1 & 2 & 3 \\0 & 2 & 3 \\1 & 0 & 1\end{array} \right]​101​220​331​​

Use matrix inversion to solve the given system of linear equations. ​ 4x+15y=1 x+4y=1 ​ x= x=\frac{\text { }}{\text { }}x= ​ ​ y= y=\frac{\text { }}{\text { }}y= ​

Find the matrix product, if possible. [3−50241]⋅[4520]\left[ \begin{array} { c c c } 3 & - 5 & 0 \\2 & 4 & 1\end{array} \right] \cdot \left[ \begin{array} { l l } 4 & 5 \\2 & 0\end{array} \right][32​−54​01​]⋅[42​50​]

Compute the determinant of the matrix. If the determinant is nonzero, use the formula for inverting a 2×22 \times 22×2 matrix to calculate the inverse of the given matrix. [122−1]\left[ \begin{array} { c c } 1 & 2 \\2 & - 1\end{array} \right][12​2−1​]

Obtain the technology matrix from the following input-output table. to from 0 200 2,100 500 300 2,100 0 0 4,200 Total Output 800 2,000 3,000 ​

If A=[3724]A = \left[ \begin{array} { l l } 3 & 7 \\2 & 4\end{array} \right]A=[32​74​] and B=[−364−2]B = \left[ \begin{array} { c c } - 3 & 6 \\4 & - 2\end{array} \right]B=[−34​6−2​] , then find A−2BA - 2 BA−2B .

Reduce the payoff matrix by dominance. B A a b c 2 -2 -6 5 0 3 3 -3 10 3 -6 -4

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Nonlinear Functions and Models

The Mathematics of Finance

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Further Integration Techniques and Applications of the Integral

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Translate the given system of equations into matrix form. $\left\{ \begin{array} { r } x + y - z = 1 \\8 x + y + z = 3 \\\frac { 3 x } { 4 } + \frac { z } { 2 } = 4\end{array} \right\}$

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

Use row reduction to find the inverse of the given matrix, if it exists, and check your answer by multiplication. $\left[ \begin{array} { l l } 2 & 2 \\8 & 8\end{array} \right]$

Reduce the payoff matrix by dominance. B p r s s a b c 1 -2 -5 4 0 2 3 -3 10 3 -5 -4

Use row reduction to find the inverse of the given matrix, if it exists. Check your answer by multiplication. $\left[ \begin{array} { l l l } 1 & 2 & 3 \\0 & 2 & 3 \\1 & 0 & 1\end{array} \right]$

Use matrix inversion to solve the given system of linear equations. 4x+15y=1 x+4y=1 $x=\frac{\text { }}{\text { }}$ $y=\frac{\text { }}{\text { }}$

Find the matrix product, if possible. $\left[ \begin{array} { c c c } 3 & - 5 & 0 \\2 & 4 & 1\end{array} \right] \cdot \left[ \begin{array} { l l } 4 & 5 \\2 & 0\end{array} \right]$

Compute the determinant of the matrix. If the determinant is nonzero, use the formula for inverting a $2 \times 2$ matrix to calculate the inverse of the given matrix. $\left[ \begin{array} { c c } 1 & 2 \\2 & - 1\end{array} \right]$

Obtain the technology matrix from the following input-output table. to from 0 200 2,100 500 300 2,100 0 0 4,200 Total Output 800 2,000 3,000

If $A = \left[ \begin{array} { l l } 3 & 7 \\2 & 4\end{array} \right]$ and $B = \left[ \begin{array} { c c } - 3 & 6 \\4 & - 2\end{array} \right]$ , then find $A - 2 B$ .