Questions: Find the production matrix for the following input-output and demand matrices using the open model. A=[ 0.1 0.3 0.45 0.1 ] D=[ 3 5 ] The production matrix is (Round the final answer to the nearest hundredth as needed. Round all intermediate values to four decimal places as needed.)

Find the production matrix for the following input-output and demand matrices using the open model.

A=[
0.1 0.3
0.45 0.1
] D=[
3
5
]

The production matrix is

(Round the final answer to the nearest hundredth as needed. Round all intermediate values to four decimal places as needed.)

Transcript text: Find the production matrix for the following input-output and demand matrices using the open model. \[ A=\left[\begin{array}{rr} 0.1 & 0.3 \\ 0.45 & 0.1 \end{array}\right] \quad D=\left[\begin{array}{l} 3 \\ 5 \end{array}\right] \] The production matrix is $\square$ $\square$ (Round the final answer to the nearest hundredth as needed. Round all intermediate values to four decimal places as needed.)

Solution

Solution Steps

To find the production matrix using the open model, we use the formula $ X = (I - A)^{-1} \cdot D $, where $ I $ is the identity matrix of the same size as $ A $. First, calculate $ I - A $, then find its inverse, and finally multiply the inverse by the demand matrix $ D $.

Step 1: Define the Matrices

We start with the input-output matrix $ A $ and the demand matrix $ D $: \[ A = \begin{bmatrix} 0.1 & 0.3 \\ 0.45 & 0.1 \end{bmatrix}, \quad D = \begin{bmatrix} 3 \\ 5 \end{bmatrix} \]

Step 2: Calculate the Identity Matrix

The identity matrix $ I $ of the same size as $ A $ is: \[ I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \]

Step 3: Compute $ I - A $

Next, we compute $ I - A $: \[ I - A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} - \begin{bmatrix} 0.1 & 0.3 \\ 0.45 & 0.1 \end{bmatrix} = \begin{bmatrix} 0.9 & -0.3 \\ -0.45 & 0.9 \end{bmatrix} \]

Step 4: Find the Inverse of $ I - A $

We then find the inverse of $ I - A $: \[ (I - A)^{-1} = \begin{bmatrix} 1.3333 & 0.4444 \\ 0.6667 & 1.3333 \end{bmatrix} \]

Step 5: Calculate the Production Matrix $ X $

Now, we calculate the production matrix $ X $ using the formula $ X = (I - A)^{-1} \cdot D $: \[ X = \begin{bmatrix} 1.3333 & 0.4444 \\ 0.6667 & 1.3333 \end{bmatrix} \cdot \begin{bmatrix} 3 \\ 5 \end{bmatrix} = \begin{bmatrix} 6.22 \\ 8.67 \end{bmatrix} \]