You are given a technology matrix \(A\) and an external demand vector \(D .\) Find the corresponding production vector \(X\). $A=\left[\begin{array}{lll}0.5 & 0.1 & 0 \\ 0.1 & 0.5 & 0.1 \\ 0 & 0 & 0.5\end{array}\right], D=\left[\begin{array}{l}3,000 \\ 3,800 \\\ 2,000\end{array}\right]$

Y o u ~ a n d ~ y o u r ~ f r i e n d ~ h a v e ~ c o m e ~ u p ~ 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.

Textbook Writing You are writing a college-level textbook on finite mathematics, and are trying to come up with the best combination of word problems. Over the years, you have accumulated a collection of amusing problems, serious applications, long complicated problems, and "generic" problems. \({ }^{25}\) Before your book is published, it must be scrutinized by several reviewers who, it seems, are never satisfied with the mix you use. You estimate that there are three kinds of reviewers: the "no-nonsense" types who prefer applications and generic problems, the "dead serious" types, who feel that a collegelevel text should be contain little or no humor and lots of long complicated problems, and the "laid-back" types, who believe that learning best takes place in a light-hearted atmosphere bordering on anarchy. You have drawn up the following chart, where the payoffs represent the reactions of reviewers on a scale of \(-10\) (ballistic) to \(+10\) (ecstatic): Reviewers ou \begin{tabular}{|l|c|c|c|} \hline & No-Nonsense & Dead Serious & Laid-Back \\ \hline Amusing & \(-5\) & \(-10\) & 10 \\ \hline Serious & 5 & 3 & 0 \\ \hline Long & \(-5\) & 5 & 3 \\ \hline Generic & 5 & 3 & \(-10\) \\ \hline \end{tabular} a. Your first draft of the book contained no generic problems, and equal numbers of the other categories. If half the reviewers of your book were "dead serious" and the rest were equally divided between the "no-nonsense" and "laid-back" types, what score would you expect? b. In your second draft of the book, you tried to balance the content by including some generic problems and eliminating several amusing ones, and wound up with a mix of which one eighth were amusing, one quarter were serious, three eighths were long, and a quarter were generic. What kind of reviewer would be least impressed by this mix? c. What kind of reviewer would be most impressed by the mix in your second draft?

Calculate (a) \(P^{2}=P \cdot P\) (b) \(P^{4}=P^{2} \cdot P^{2}\) and \(\left(\right.\) c) \(P^{8} .\) Round all entries to four decimal places.) (d) Without computing it explicitly, find \(P^{1000}\). $$ P=\left[\begin{array}{lll} 0.3 & 0.3 & 0.4 \\ 0.3 & 0.3 & 0.4 \\ 0.3 & 0.3 & 0.4 \end{array}\right] $$

Reduce the payoff matrices by dominance. $$ \begin{aligned} &\mathbf{B}\\\ &\left.\begin{array}{l} a & b & c \\ p \\ q \\ r \\ s & 2 & -4 & 9 \\ 1 & 1 & 0 \\ -1 & -2 & -3 \\ 1 & 1 & -1 \end{array}\right] \end{aligned} $$

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.