Exam 5: Matrix Algebra and Applications

Given $A = \left[ \begin{array} { c c } 7 & 2 \\- 9 & 6\end{array} \right]$ , $B = \left[ \begin{array} { l l } 3 & 5\end{array} \right]$ , and $C = \left[ \begin{array} { c c c } - 1 & 4 & 8 \\2 & 0 & 3\end{array} \right]$ , which of the following can be calculated

Compute the product, if possible. $\left[ \begin{array} { l l } 2 & - 4 \\2 & - 4\end{array} \right] \cdot \left[ \begin{array} { l l } 4 & - 4 \\1 & - 7\end{array} \right]$

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

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

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 .

If $A = \left[ \begin{array} { l l } 4 & 6 \\1 & 2\end{array} \right]$ , and $B = \left[ \begin{array} { c c } - 1 & 4 \\5 & - 2\end{array} \right]$ then find $A - 2 B$ .

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

Decide whether the game is strictly determined. B pq a b 1 1 4 -3 __________ (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 __________

Given the technology matrix A, and an external demand vector D, find the production vector X. ​ $A = \left[ \begin{array} { c c c } 0.5 & 0.1 & 0 \\0 & 0.5 & 0.1 \\0 & 0 & 0.5\end{array} \right]$ , $D = \left[ \begin{array} { l } 2200 \\4900 \\3400\end{array} \right]$

Obtain the technology matrix from the input-output table. to A B C from 0 200 300 500 400 300 0 0 600 Total Output 1,000 2,000 3,000

Let A be the technology matrix, where Sector 1 = wood, and Sector 2 = paper. ​ $A = \left[ \begin{array} { l l } 0.2 & 0.06 \\0.4 & 0.01\end{array} \right]$ ​ __________ 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.

Let $A = \left[ \begin{array} { c c c c } 7 & - 5 & 5 & - 7 \\8 & 4 & 0 & 0\end{array} \right]$ and $B = \left[ \begin{array} { c c c c } 4 & 5 & - 4 & - 3 \\2 & 5 & 0 & - 3\end{array} \right]$ ​ Evaluate A + B.

The following tables give annual production costs and profits at Gauss Jordan Sneakers. Production costs Gauss Grip \ 1,100 \ 2,700 \ 3,500 Air Gauss \ 1,000 \ 1,800 \ 2,900 Gauss Gel \ 1,900 \ 2,800 \ 1,200 Profits Gauss Grip \ 17,000 \ 13,000 \ 24,000 Air Gauss \ 8,000 \ 16,000 \ 13,000 Gauss Gel \ 6,000 \ 12,000 \ 20,000 Write the matrix algebraic formula to compute the revenues from each sector each year.

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): To Equipment Components From Equipment 8,000 500 Components 26,000 34,000 Total output 82,000 121,000 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.

Compute the product of the two matrices (if possible). $\left[ \begin{array} { c c } 4 & - 1\end{array} \right] \cdot \left[ \begin{array} { c c c c } 2 & 0 & 3 & 5 \\21 & - 1 & 7 & - 3\end{array} \right]$

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.

Decide whether the game is strictly determined. If it is, give the players' optimal pure strategies and the value of the game. B pqr a b -3 2 -4 -3 3 -3

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) To 1 2 from 79 1,094 0 1,233 64,859 13,086 7 1,074 0 0 21,787 0 0 0 0 1,375 Total Output 230,677 135,110 129,374 44,140 Determine how these four sectors would react to an increase in demand for Sector 1 production of $1000 million.

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): To Equipment Components From Equipment 8,000 700 Components 21,000 24,000 Total output 90,000 141,000 ​ 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. ​

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