# Chapter 4: Chapter 4

Problem 8

Either a mixed column or mixed row strategy is given. In each case, use $$ P=\left[\begin{array}{rrr} 0 & -1 & 5 \\ 2 & -2 & 4 \\ 0 & 3 & 0 \\ 1 & 0 & -5 \end{array}\right] $$ and find the optimal pure strategy (or strategies) the other player should use. Express the answer as a row or column matrix. Also determine the resulting expected payoff. [HINT: See Example 2.] $$ R=\left[\begin{array}{llll} 0.8 & 0.2 & 0 & 0 \end{array}\right] $$

Problem 8

You are given a technology matrix A and an external demand vector \(D\). Find the corresponding production vector \(X .\) [HINT: See Quick Example \(1 .]\) $$ A=\left[\begin{array}{ll} 0.1 & 0.2 \\ 0.4 & 0.5 \end{array}\right], D=\left[\begin{array}{l} 24,000 \\ 14,000 \end{array}\right] $$

Problem 8

Use row reduction to find the inverses of the given matrices if they exist, and check your answers by multiplication. $$ \left[\begin{array}{ll} 0 & 1 \\ 1 & 1 \end{array}\right] $$

Problem 8

In Exercises \(1-10,\) find the dimensions of the given matrix, and identify the given entry. $$ C=\left[\begin{array}{cccc} x & y & w & e \\ z & t+1 & 3 & 0 \end{array}\right] ; C_{23} $$

Problem 81

Comment on the following claim: Every matrix equation represents a system of equations.

Problem 82

When is it true that both \(A B\) and \(B A\) are defined, even though neither \(A\) nor \(B\) is a square matrix?

Problem 84

Make up an application whose solution reads as follows: "Total revenue $=\left[\begin{array}{lll}10 & 100 & 30\end{array}\right]\left[\begin{array}{rrr}10 & 0 & 3 \\ 1 & 2 & 0 \\ 0 & 1 & 40\end{array}\right],$

Problem 86

Define the naive product \(A \square B\) of two \(m \times n\) matrices \(A\) and \(B\) by $$ (A \square B)_{i j}=A_{i j} B_{i j} $$ (This is how someone who has never seen matrix multiplication before might think to multiply matrices.) Referring to Example 1 in this section, compute and comment on the meaning of \(P \square\left(Q^{T}\right)\).

Problem 9

Reduce the given payoff matrix by dominance. $$ \left.\begin{array}{rrr} p & q & r \\ a & 1 & 1 & 10 \\ b & 2 & 3 & -4 \end{array}\right] $$

Problem 9

Use row reduction to find the inverses of the given matrices if they exist, and check your answers by multiplication. $$ \left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right] $$