Problem

Find the saddle points and values of the payolf matrices. (To enter the saddle points, order your answers from smallest to largest $r$, then from smallest to largest $\mathrm{c}$.)
(a)
\[
\left[\begin{array}{rrr}
4 & 4 & 8 \\
1 & -2 & -3 \\
4 & 4 & 9
\end{array}\right]
\]
saddle points
\[
\begin{array}{l}
(r, c)=(\square) \\
(r, c)=(\square) \\
(r, c)=(\square) \\
(r, c)=(\square)
\end{array}
\]
value
(b) $\left[\begin{array}{rrrr}6 & 7 & 6 & 8 \\ 2 & 6 & 1 & 16 \\ 6 & 9 & 6 & 11\end{array}\right]$
sacolte points
\[
\begin{array}{l}
(r, c)=(\square) \\
(b, c)=(\square) \\
(r, c)=(\square) \\
(r, c)=(\square)
\end{array}
\]
value

Answer

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Answer

So, the final answer is \(\boxed{\text{Saddle points are (1,1), (1,2) with value 4, (3,3) with value 9 for the first matrix and (1,1) with value 6 for the second matrix.}}\)

Steps

Step 1 :First, find the minimum of each row in the matrix.

Step 2 :For the first row, the minimum is \(4\).

Step 3 :For the second row, the minimum is \(-3\).

Step 4 :For the third row, the minimum is \(4\).

Step 5 :Next, find the maximum of each column in the matrix.

Step 6 :For the first column, the maximum is \(4\).

Step 7 :For the second column, the maximum is \(4\).

Step 8 :For the third column, the maximum is \(9\).

Step 9 :The intersection of the row minimum and column maximum gives us the saddle points. In this case, we have two saddle points: \((1,1)\) and \((1,2)\) where the value is \(4\), and \((3,3)\) where the value is \(9\).

Step 10 :For the second part, again find the minimum of each row in the matrix.

Step 11 :For the first row, the minimum is \(6\).

Step 12 :For the second row, the minimum is \(1\).

Step 13 :For the third row, the minimum is \(6\).

Step 14 :Then, find the maximum of each column in the matrix.

Step 15 :For the first column, the maximum is \(6\).

Step 16 :For the second column, the maximum is \(9\).

Step 17 :For the third column, the maximum is \(6\).

Step 18 :For the fourth column, the maximum is \(16\).

Step 19 :The intersection of the row minimum and column maximum gives us the saddle points. In this case, we have one saddle point: \((1,1)\) where the value is \(6\).

Step 20 :So, the final answer is \(\boxed{\text{Saddle points are (1,1), (1,2) with value 4, (3,3) with value 9 for the first matrix and (1,1) with value 6 for the second matrix.}}\)

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