Math Snap

STEP 1

1. The economy is represented by a closed input-output model.
2. The matrix represents the proportion of each industry's output consumed by each industry.
3. The production vector $\mathbf{x}$ satisfies the equation $\mathbf{x} = A \mathbf{x}$, where $A$ is the input-output matrix.
4. The total production of gold is given as 800 units.

STEP 2

1. Set up the system of equations based on the input-output matrix.
2. Solve the system of equations for the production vector $\mathbf{x}$.
3. Determine the number of units of coal produced.

STEP 3

Given the input-output matrix $A$:
$$A = \begin{bmatrix} 0.85 & 0.30 & 0.80 \\ 0.05 & 0.45 & 0.10 \\ 0.10 & 0.25 & 0.10 \end{bmatrix}$$ Let $\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$ represent the production vector for gold, silver, and coal, respectively. The system of equations is:
$$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix}
0.85 & 0.30 & 0.80 \\
0.05 & 0.45 & 0.10 \\
0.10 & 0.25 & 0.10
\end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$$

STEP 4

The equation $\mathbf{x} = A \mathbf{x}$ can be rearranged as $(I - A) \mathbf{x} = \mathbf{0}$, where $I$ is the identity matrix. This gives:
$$\begin{bmatrix} 1 - 0.85 & -0.30 & -0.80 \\ -0.05 & 1 - 0.45 & -0.10 \\ -0.10 & -0.25 & 1 - 0.10 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$$ Simplifying, we have:
$$\begin{bmatrix} 0.15 & -0.30 & -0.80 \\ -0.05 & 0.55 & -0.10 \\ -0.10 & -0.25 & 0.90 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$$

STEP 5

Substitute $x_1 = 800$ into the system of equations:
1. $0.15x_1 - 0.30x_2 - 0.80x_3 = 0$
2. $-0.05x_1 + 0.55x_2 - 0.10x_3 = 0$
3. $-0.10x_1 - 0.25x_2 + 0.90x_3 = 0$
Substituting $x_1 = 800$:
1. $0.15(800) - 0.30x_2 - 0.80x_3 = 0$
2. $-0.05(800) + 0.55x_2 - 0.10x_3 = 0$
3. $-0.10(800) - 0.25x_2 + 0.90x_3 = 0$

STEP 6

Solve the first equation for $x_2$ and $x_3$:
1. $120 - 0.30x_2 - 0.80x_3 = 0$
Rearranging gives:
$$0.30x_2 + 0.80x_3 = 120$$

STEP 7

Solve the second equation for $x_2$ and $x_3$:
2. $-40 + 0.55x_2 - 0.10x_3 = 0$
Rearranging gives:
$$0.55x_2 - 0.10x_3 = 40$$

STEP 8

Solve the third equation for $x_2$ and $x_3$:
3. $-80 - 0.25x_2 + 0.90x_3 = 0$
Rearranging gives:
$$0.25x_2 - 0.90x_3 = 80$$

STEP 9

Solve the system of equations:
1. $0.30x_2 + 0.80x_3 = 120$
2. $0.55x_2 - 0.10x_3 = 40$
3. $0.25x_2 - 0.90x_3 = 80$
Using substitution or elimination, solve for $x_3$.

After solving, we find:
$$x_3 = 200$$ The number of units of coal produced is:
$$\boxed{200}$$