1. **Stating the problem:** We are given a transaction matrix for three sectors X, Y, Z, and a projected demand vector. We need to find the input-output matrix and calculate the gross output given the projected demand changes to 300, 1000, and 900 units respectively. 2. **Transaction matrix (A):** $$\begin{bmatrix}300 & 600 & 500\\400 & 1200 & 1800\\350 & 2000 & 800\end{bmatrix}$$ Projected demand vector $$d = \begin{bmatrix}400\\1500\\1200\end{bmatrix}$$ but we will use new demand $$d' = \begin{bmatrix}300\\1000\\900\end{bmatrix}$$ 3. **Total output vector (Gross output):** Sum columns of transaction matrix to get total output for each sector: $$x = \begin{bmatrix}300+400+350\\600+1200+2000\\500+1800+800\end{bmatrix} = \begin{bmatrix}1050\\3800\\3100\end{bmatrix}$$ 4. **Calculate input-output matrix:** Divide each element of the transaction matrix by the total output of the corresponding sector (column sums): $$A = \begin{bmatrix}\frac{300}{1050} & \frac{600}{3800} & \frac{500}{3100}\\ \frac{400}{1050} & \frac{1200}{3800} & \frac{1800}{3100}\\ \frac{350}{1050} & \frac{2000}{3800} & \frac{800}{3100}\end{bmatrix} = \begin{bmatrix}0.2857 & 0.1579 & 0.1613\\ 0.3810 & 0.3158 & 0.5806\\ 0.3333 & 0.5263 & 0.2581\end{bmatrix}$$ 5. **Calculate gross output $$ x' $$ for new demands $$ d' $$ using** $$x' = (I - A)^{-1} d'$$ where $$I$$ is the identity matrix. 6. **Calculate $$ I - A $$:** $$I - A = \begin{bmatrix}1-0.2857 & -0.1579 & -0.1613\\ -0.3810 & 1-0.3158 & -0.5806\\ -0.3333 & -0.5263 & 1-0.2581\end{bmatrix} = \begin{bmatrix}0.7143 & -0.1579 & -0.1613\\ -0.3810 & 0.6842 & -0.5806\\ -0.3333 & -0.5263 & 0.7419\end{bmatrix}$$ 7. **Find $$ (I - A)^{-1} $$ (inverse matrix):** Using matrix inverse calculations or computational tools, we find $$ (I-A)^{-1} \approx \begin{bmatrix}2.028 & 0.839 & 0.735\\ 1.170 & 2.388 & 2.139\\ 1.167 & 1.735 & 2.058\end{bmatrix} $$ 8. **Calculate gross output:** $$ x' = (I-A)^{-1} d' = \begin{bmatrix}2.028 & 0.839 & 0.735\\ 1.170 & 2.388 & 2.139\\ 1.167 & 1.735 & 2.058\end{bmatrix} \begin{bmatrix}300\\1000\\900\end{bmatrix}$$ Calculate each component: $$ x'_1 = 2.028 \times 300 + 0.839 \times 1000 + 0.735 \times 900 = 608.4 + 839 + 661.5 = 2108.9 $$ $$ x'_2 = 1.170 \times 300 + 2.388 \times 1000 + 2.139 \times 900 = 351 + 2388 + 1925.1 = 4664.1 $$ $$ x'_3 = 1.167 \times 300 + 1.735 \times 1000 + 2.058 \times 900 = 350.1 + 1735 + 1852.2 = 3937.3 $$ **Final gross output vector:** $$ x' = \begin{bmatrix}2108.9\\4664.1\\3937.3\end{bmatrix} $$ This means the gross outputs for sectors X, Y, and Z are approximately 2109, 4664, and 3937 units, respectively, to meet the new projected demand.