1. **State the problem:** We have an economy with three sectors: agriculture, manufacturing, and energy. Each sector requires inputs from all three sectors to produce one dollar's worth of output. We want to find the total output of each sector needed to satisfy a final demand of $10 billion for agriculture, $15 billion for manufacturing, and $20 billion for energy. 2. **Set up the input-output matrix and final demand vector:** Let $x_a$, $x_m$, and $x_e$ be the total outputs of agriculture, manufacturing, and energy respectively. The input coefficients matrix $A$ is: $$ A = \begin{bmatrix} 0.20 & 0.20 & 0.20 \\ 0.40 & 0.10 & 0.10 \\ 0.30 & 0.10 & 0.10 \end{bmatrix} $$ The final demand vector $d$ is: $$ d = \begin{bmatrix}10 \\ 15 \\ 20\end{bmatrix} \text{ (in billions)} $$ 3. **Use the Leontief input-output model formula:** $$ x = Ax + d $$ which rearranges to $$ (I - A)x = d $$ where $I$ is the identity matrix. 4. **Calculate $(I - A)$:** $$ I - A = \begin{bmatrix} 1-0.20 & -0.20 & -0.20 \\ -0.40 & 1-0.10 & -0.10 \\ -0.30 & -0.10 & 1-0.10 \end{bmatrix} = \begin{bmatrix} 0.80 & -0.20 & -0.20 \\ -0.40 & 0.90 & -0.10 \\ -0.30 & -0.10 & 0.90 \end{bmatrix} $$ 5. **Solve the system $(I - A)x = d$:** We want to find $x$ such that $$ \begin{bmatrix} 0.80 & -0.20 & -0.20 \\ -0.40 & 0.90 & -0.10 \\ -0.30 & -0.10 & 0.90 \end{bmatrix} \begin{bmatrix}x_a \\ x_m \\ x_e\end{bmatrix} = \begin{bmatrix}10 \\ 15 \\ 20\end{bmatrix} $$ 6. **Use matrix inversion or substitution to solve:** Calculate the inverse of $(I - A)$ (denote it as $M^{-1}$) and then compute $$ x = M^{-1} d $$ 7. **Intermediate calculations (using a calculator or software):** The inverse matrix is approximately: $$ M^{-1} \approx \begin{bmatrix} 1.818 & 0.545 & 0.455 \\ 0.909 & 1.364 & 0.136 \\ 0.682 & 0.227 & 1.136 \end{bmatrix} $$ Multiply $M^{-1}$ by $d$: $$ x = \begin{bmatrix} 1.818 & 0.545 & 0.455 \\ 0.909 & 1.364 & 0.136 \\ 0.682 & 0.227 & 1.136 \end{bmatrix} \begin{bmatrix}10 \\ 15 \\ 20\end{bmatrix} = \begin{bmatrix} 1.818 \times 10 + 0.545 \times 15 + 0.455 \times 20 \\ 0.909 \times 10 + 1.364 \times 15 + 0.136 \times 20 \\ 0.682 \times 10 + 0.227 \times 15 + 1.136 \times 20 \end{bmatrix} $$ Calculate each component: - $x_a = 18.18 + 8.175 + 9.10 = 35.455$ - $x_m = 9.09 + 20.46 + 2.72 = 32.27$ - $x_e = 6.82 + 3.405 + 22.72 = 32.945$ 8. **Final answer:** The total outputs needed (in billions) are approximately: $$ x_a = 35.46, \quad x_m = 32.27, \quad x_e = 32.95 $$ This means to satisfy the final demand, agriculture must produce about 35.46 billion, manufacturing about 32.27 billion, and energy about 32.95 billion dollars worth of output.