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 50 billion for agriculture, 27 billion for manufacturing, and 46 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. Input coefficients matrix $A$: $$ A = \begin{bmatrix} 0.30 & 0.30 & 0.20 \\ 0.20 & 0.20 & 0.30 \\ 0.20 & 0.40 & 0.40 \end{bmatrix} $$ Final demand vector $d$: $$ d = \begin{bmatrix}50 \\ 27 \\ 46\end{bmatrix} $$ 3. **Formulate the system:** The total output vector $x$ satisfies: $$ x = Ax + d$$ which can be rearranged to: $$ (I - A)x = d $$ where $I$ is the identity matrix. 4. **Calculate $I - A$:** $$ I - A = \begin{bmatrix} 1-0.30 & -0.30 & -0.20 \\ -0.20 & 1-0.20 & -0.30 \\ -0.20 & -0.40 & 1-0.40 \end{bmatrix} = \begin{bmatrix} 0.70 & -0.30 & -0.20 \\ -0.20 & 0.80 & -0.30 \\ -0.20 & -0.40 & 0.60 \end{bmatrix} $$ 5. **Solve the system $(I - A)x = d$:** We solve for $x$ by computing: $$ x = (I - A)^{-1} d $$ Using matrix inversion and multiplication (rounded to six decimals): Inverse matrix $(I - A)^{-1}$ approximately: $$ \begin{bmatrix} 1.818182 & 0.818182 & 0.636364 \\ 0.545455 & 1.363636 & 0.818182 \\ 0.727273 & 0.909091 & 1.818182 \end{bmatrix} $$ Multiply by $d$: $$ x = \begin{bmatrix} 1.818182 & 0.818182 & 0.636364 \\ 0.545455 & 1.363636 & 0.818182 \\ 0.727273 & 0.909091 & 1.818182 \end{bmatrix} \begin{bmatrix}50 \\ 27 \\ 46\end{bmatrix} = \begin{bmatrix} 1.818182 \times 50 + 0.818182 \times 27 + 0.636364 \times 46 \\ 0.545455 \times 50 + 1.363636 \times 27 + 0.818182 \times 46 \\ 0.727273 \times 50 + 0.909091 \times 27 + 1.818182 \times 46 \end{bmatrix} $$ Calculate each component: - Agriculture output: $$ = 90.9091 + 22.0909 + 29.2727 = 142.2737 $$ 6. **Final answer:** The output of the agricultural sector needed is approximately **142.274 billion dollars** (rounded to three decimal places).