Find the production matrix for the input-output and demand matrices using the open model.
\[
A=\left[\begin{array}{rrr}
0 & 0.25 & 0.33 \\
0.50 & 0 & 0.25 \\
0.25 & 0.25 & 0
\end{array}\right], D=\left[\begin{array}{l}
531 \\
268 \\
112
\end{array}\right]
\]

Step 1 ：We are given the input-output matrix A and the demand matrix D as follows:

Step 2 ：\[A=\left[\begin{array}{rrr} 0 & 0.25 & 0.33 \\ 0.50 & 0 & 0.25 \\ 0.25 & 0.25 & 0 \end{array}\right], D=\left[\begin{array}{l} 531 \\ 268 \\ 112 \end{array}\right]\]

Step 3 ：We are asked to find the production matrix X using the open model. The formula for this is X = (I - A)^{-1} * D, where I is the identity matrix.

Step 4 ：First, we calculate I - A. The identity matrix I is a matrix with ones on the diagonal and zeros elsewhere. In this case, I is a 3x3 matrix, so we have:

Step 5 ：\[I = \left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\]

Step 6 ：Subtracting A from I gives us:

Step 7 ：\[I - A = \left[\begin{array}{rrr} 1 & -0.25 & -0.33 \\ -0.50 & 1 & -0.25 \\ -0.25 & -0.25 & 1 \end{array}\right]\]

Step 8 ：Next, we need to find the inverse of this matrix, denoted as (I - A)^{-1}.

Step 9 ：Finally, we multiply this inverse matrix by the demand matrix D to find the production matrix X.

Step 10 ：Doing this gives us the production matrix X as follows:

Step 11 ：\[X=\left[\begin{array}{l} 937.24419684 \\ 878.07985144 \\ 565.83101207 \end{array}\right]\]

Step 12 ：So, the final answer is \(\boxed{X=\left[\begin{array}{l} 937.24419684 \\ 878.07985144 \\ 565.83101207 \end{array}\right]}\)