Find the production matrix for the following input-output and demand matrices \[ A=\left[\begin{array}{rrr} 0.2 & 0 & 0.08 \\ 0 & 0.8 & 0.02 \\ 0 & 0.2 & 0.88 \end{array}\right] \quad D=\left[\begin{array}{l} 2 \\ 1 \\ 2 \end{array}\right] \] 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 \(\boxed{X=\left[\begin{array}{l} 2.5 \\ 1.25 \\ 2.25 \end{array}\right]}\)