91Ó°ÊÓ

Let \(A\) be the technology matrix \(A=\left[\begin{array}{ll}0.2 & 0.05 \\ 0.8 & 0.01\end{array}\right]\), where Sector 1 is paper and Sector 2 is wood. Fill in the missing quantities. a. units of wood are needed to produce one unit of paper. b. units of paper are used in the production of one unit of paper. c. The production of each unit of wood requires the use of units of paper.

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

You are given a technology matrix \(A\) and an external demand vector \(D .\) Find the corresponding production vector \(X\). \(A=\left[\begin{array}{ll}0.5 & 0.4 \\ 0 & 0.5\end{array}\right], D=\left[\begin{array}{l}10,000 \\ 20,000\end{array}\right]\)

Can an external demand be met by an economy whose technology matrix \(A\) is the identity matrix? Explain.

More Retail Discount Wars Your Abercrom B men's fashion outlet has a \(30 \%\) chance of launching an expensive new line of used auto-mechanic dungarees (complete with grease stains) and a \(70 \%\) 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 \(20 \%\) chance that Abercrom A will opt for the empty shirt option. The following payoff matrix gives the number of customers your outlet can expect to gain from Abercrom A in each situation: Abercrom \(\mathbf{A}\) \(\begin{array}{ll}\text { Torn Shirts } & \text { Empty Shirts }\end{array}\) Mechanics \(\left[\begin{array}{rr}10 & -40 \\ -30 & 50\end{array}\right]\) Abercrom B \(\quad\) Military What is the expected resulting effect on your customer base?

Resource Allocation The Arctic Juice Company makes three juice blends: PineOrange, using 2 quarts of pineapple juice and 2 quarts of orange juice per gallon; PineKiwi, using 3 quarts of pineapple juice and 1 quart of kiwi juice per gallon; and OrangeKiwi, using 3 quarts of orange juice and 1 quart of kiwi juice per gallon. The amount of each kind of juice the company has on hand varies from day to day. How many gallons of each blend can it make on a day with the following stocks? a. 800 quarts of pineapple juice, 650 quarts of orange juice, 350 quarts of kiwi juice. b. 650 quarts of pineapple juice, 800 quarts of orange juice, 350 quarts of kiwi juice. c. \(A\) quarts of pineapple juice, \(B\) quarts of orange juice, \(C\) quarts of kiwi juice.

Multiple Choice: If \(A\) and \(B\) are square matrices with \(A B=I\) and \(B A=I\), then (A) \(B\) is the inverse of \(A\). (B) \(A\) and \(B\) must be equal. (C) \(A\) and \(B\) must both be singular. (D) At least one of \(A\) and \(B\) is singular.