Exam 5: Matrix Algebra and Applications

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

Use row reduction to find the inverse of the given matrix, if it exists, and check your answer by multiplication. $\left[ \begin{array} { c c c } 1 & 4 & - 1 \\4 & 5 & - 4 \\- 1 & 4 & 5\end{array} \right]$

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

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 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. I II III 170 120 130 130 220 140 130 120 220 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

Given the following matrix. $C = \left[ \begin{array} { c c } - 2 & 4 \\2 & 6 \\- 7 & - 2 \\0 & 9\end{array} \right]$ What are the dimensions $\_\_\_\_\_\_\_\text { x }\_\_\_\_\_\_\_$ What is the value of the element $C _ { 32 }$ $c _ { 32 } =\_\_\_\_\_\_\_$

Solve for x, y, and z. $\left[ \begin{array} { c c } x - y & 5 \\z & x + y\end{array} \right] = \left[ \begin{array} { c c } - 2 & 5 \\3 & 8\end{array} \right]$

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: Japanese Commander's Strategies NorthernRoute SouthernRoute Kenney's Northern Route 2 2.5 Strategies Split Reconnaissance 0.5 3 Southern Route 1 3.5 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

Calculate the expected value of the game with the given payoff matrix using the mixed strategy supplied. ​ ​ $P = \left[ \begin{array} { c c c c } 2 & 0 & - 1 & 2 \\- 1 & 0 & 0 & - 2 \\- 2 & 0 & 0 & 1 \\3 & 1 & - 1 & 1\end{array} \right]$ , $R = [ 0.5,0,0,0.5 ]$ , $C = [ 1,0,0,0 ] ^ { T }$

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 40 10 Football Club 10 15 Total Output 100 200 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 __________

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. $P = \left[ \begin{array} { c c } - 7 & - 8 \\- 8 & 7\end{array} \right]$

Use matrix inversion to solve the system of linear equations. ​ x+2y+z =17 x+y+z =10 3x+y+z =4 ​ $x=\frac{\text { }}{\text { }}$ ​ $y=\frac{\text { }}{\text { }}$ ​ $z=\frac{\text { }}{\text { }}$

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.

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 3 4 From 171 30.6 120.2 14.2 0 190.1 55.7 12.6 2.9 40.2 1,418.1 1,242 0.1 7.2 40.3 326.4 Total Output 3,278.9 2,189 6,541.3 4,065.3 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.

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.

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.

Find $A ^ { 80 }$ when $A = \left[ \begin{array} { l l l } 0 & 1 & 1 \\0 & 0 & 1 \\0 & 0 & 0\end{array} \right]$ .

In January, the Left Coast Bookstore chain sold 600 hardcover books, 1,100 softcover books, and 1,700 plastic books in San Francisco; it sold 300 hardcover, 100 softcover, and 200 plastic books in Los Angeles. Hardcover books sell for $26 each, softcover books sell for $8 each, and plastic books sell for $15 each. Hard Soft Plastic San Francisco 600 1,100 1,700 Los Angeles 300 100 200 Suppose that each hardcover book costs the stores $10, each softcover book costs $5, and each plastic book costs $10. Use matrix operations to compute the the total profit at each store in January.

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. $P = \left[ \begin{array} { l l } - 7 & - 2 \\- 2 & - 3\end{array} \right]$

Let $A = \left[ \begin{array} { c c c } 0 & 1 & 2 \\- 3 & 5 & 6\end{array} \right]$ , $B = \left[ \begin{array} { c c } 5 & 8 \\- 2 & 0 \\7 & - 2\end{array} \right]$ , and $C = \left[ \begin{array} { c c } 10 & 5 \\2 & - 8 \\12 & 5\end{array} \right]$ . Evaluate $\left( A ^ { T } + B \right) - C$ .