Find the production matrix for the following input-output and demand matrices
$$A=\left[\begin{array}{rrr}0.2& 0& 0.08\\ 0& 0.8& 0.02\\ 0& 0.2& 0.88\end{array}\right]D=\left[\begin{array}{l}2\\ 1\\ 2\end{array}\right]$$
The production matrix is

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

Step 2 ：$$A=\left[\begin{array}{rrr}0.2& 0& 0.08\\ 0& 0.8& 0.02\\ 0& 0.2& 0.88\end{array}\right],D=\left[\begin{array}{l}2\\ 1\\ 2\end{array}\right]$$

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 ：$$I-A=\left[\begin{array}{rrr}0.8& 0& -0.08\\ 0& 0.2& -0.02\\ 0& -0.2& 0.12\end{array}\right]$$

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 ：$$X=\left[\begin{array}{l}2.5\\ 1.25\\ 2.25\end{array}\right]$$

Step 12 ：So, the final answer is $\boxed{{\displaystyle X=\left[\begin{array}{l}2.5\\ 1.25\\ 2.25\end{array}\right]}}$