Question

Find the production matrix for the following input-output and demand matrices using the open model.0.1 0.34A=D=0.45 0.451.9The production matrix is3.8(Round the final answer to the nearest hundredth as needed. Round all intermediate values to four decimal places as needed.)

Answer 1

Let the production matrix = X

`X\text{ = }(I-A)^(-1)D`

Next

`\begin{gathered} A\text{ = }\begin{bmatrix}{0.1} & {} & {0.2} \\ {} & {} & {} \\ {0.45} & {} & {0.6}\end{bmatrix} \\ D\text{ = }\begin{bmatrix}{2} & {} & {} \\ {} & {} & {} \\ {4} & & \end{bmatrix} \end{gathered}`
`I\text{ = }\begin{bmatrix}{1} & {} & {0} \\ {} & {} & {} \\ {0} & {} & {1}\end{bmatrix}`

Next, evaluate X

`\begin{gathered} I\text{ - A = }\begin{bmatrix}{1} & {} & {0} \\ {} & {} & {} \\ {0} & {} & {1}\end{bmatrix}\text{ - }\begin{bmatrix}{0.1} & {} & {0.2} \\ {} & {} & {} \\ {0.45} & {} & {0.6}\end{bmatrix} \\ =\text{ }\begin{bmatrix}{0.9} & {} & {-0.2} \\ {} & {} & {} \\ {-0.45} & {} & {0.4}\end{bmatrix} \end{gathered}`

Next, you find the inverse of I - A which is adjount of I - A divided by it determinant.

Determinant of (I - A) = 0.9 x 0.4 - (-0.45 x -0.2) = 0.36 - 0.09 = 0.27

Adjount of (I - A) is the transpose of it co-factor

`\begin{gathered} Co-\text{factor of (I - A) = }\begin{bmatrix}{0.4} & {} & {0.45} \\ {} & {} & {} \\ {0.2} & {} & {0.9}\end{bmatrix} \\ \text{Adjount = }\begin{bmatrix}{0.4} & {} & {0.2} \\ {} & {} & {} \\ {0.45} & {} & {0.9}\end{bmatrix} \end{gathered}`

Therefore

`\begin{gathered} (I-A)^(-1)\text{ = }(1)/(0.27)\begin{bmatrix}{0.4} & {} & {0.2} \\ {} & {} & {} \\ {0.45} & {} & {0.9}\end{bmatrix} \\ =\text{ }\begin{bmatrix}{1.48} & {} & {0.74} \\ {} & {} & {} \\ {1.67} & {} & {3.33}\end{bmatrix} \end{gathered}`

Now, we find x

`\begin{gathered} X=(I-A)^(-1)D \\ =\text{ }\begin{bmatrix}{1.48} & {} & {0.74} \\ {} & {} & {} \\ {1.67} & {} & {3.33}\end{bmatrix}\text{ X }\begin{bmatrix}{} & {2} & {} \\ {} & {} & {} \\ {} & {4} & {}\end{bmatrix} \\ X=\text{ }\begin{bmatrix}{} & {5.92} & {} \\ {} & {} & {} \\ {} & {16.66} & {}\end{bmatrix} \end{gathered}`
`\text{The production matrix = }\begin{bmatrix}{} & {5.92} & {} \\ {} & {} & {} \\ {} & {16.66} & {}\end{bmatrix}`