To solve this problem, we need to calculate $(I - M)^{-1}$ and then use it to find the output vector $X$ that satisfies the equation $X = (I - M)^{-1}D$.

1. Define the matrices:

   The technology matrix $M$ is: $M = \begin{bmatrix} 0.6 & 0.7 \\ 0.1 & 0.2 \end{bmatrix}$

   The identity matrix $I$ for a 2x2 system is: $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

   The final demand matrix $D$ is: $D = \begin{bmatrix} 6 \\ 4 \end{bmatrix}$

2. Calculate $I - M$:

   $I - M = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} - \begin{bmatrix} 0.6 & 0.7 \\ 0.1 & 0.2 \end{bmatrix} = \begin{bmatrix} 0.4 & -0.7 \\ -0.1 & 0.8 \end{bmatrix}$

3. Check if $(I - M)$ is invertible:

   To confirm if $(I - M)$ is invertible, calculate its determinant: $\text{det}(I - M) = (0.4)(0.8) - (-0.7)(-0.1) = 0.32 - 0.07 = 0.25$

   Since the determinant is non-zero, $(I - M)$ is invertible.

4. Calculate $(I - M)^{-1}$:

   For a 2x2 matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the inverse is given by: $\frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$

   Applying this formula: $(I - M)^{-1} = \frac{1}{0.25} \begin{bmatrix} 0.8 & 0.7 \\ 0.1 & 0.4 \end{bmatrix} = \begin{bmatrix} 3.2 & 2.8 \\ 0.4 & 1.6 \end{bmatrix}$

5. Find the output vector $X$:

   Now, we calculate $X = (I - M)^{-1}D$: $X = \begin{bmatrix} 3.2 & 2.8 \\ 0.4 & 1.6 \end{bmatrix} \begin{bmatrix} 6 \\ 4 \end{bmatrix}$

   Perform the matrix multiplication: $X = \begin{bmatrix} (3.2)(6) + (2.8)(4) \\ (0.4)(6) + (1.6)(4) \end{bmatrix} = \begin{bmatrix} 19.2 + 11.2 \\ 2.4 + 6.4 \end{bmatrix} = \begin{bmatrix} 30.4 \\ 8.8 \end{bmatrix}$

So, the output vector $X$ that satisfies the final demand $D$ is: $X = \begin{bmatrix} 30.4 \\ 8.8 \end{bmatrix}$

Answer

1. $(I - M)^{-1} = \begin{bmatrix} 3.2 & 2.8 \\ 0.4 & 1.6 \end{bmatrix}$
2. Output vector $X = \begin{bmatrix} 30.4 \\ 8.8 \end{bmatrix}$

Would you like further clarification on any steps or concepts?

Related Questions:

1. What does the technology matrix represent in an economic context?
2. How does the inverse matrix help in solving input-output models?
3. Why do we check the invertibility of $(I - M)$ before proceeding?
4. How would a change in the final demand vector $D$ affect the output vector $X$?
5. What would happen if the determinant of $(I - M)$ were zero?

Tip:

In input-output models, each element in the technology matrix represents the proportion of input from one sector required by another, providing insights into interdependencies between industries.