Diagonalization & Matrix Powers · Challenge problem

An economy has two sectors; producing one unit of sector $j$'s output requires $A_{ij}$ units of input from sector $i$, with $A = \begin{pmatrix}0.2 & 0.3\\0.1 & 0.4\end{pmatrix}$. To meet an external demand of $d = \begin{pmatrix}100\\50\end{pmatrix}$, total output must satisfy $\mathbf x = A\mathbf x + \mathbf d$, so $\mathbf x = (I-A)^{-1}\mathbf d$. Diagonalizing $A$ (whose eigenvalues are $0.5$ and $0.1$) makes $(I-A)^{-1}$ easy to build. Find the total output of sector 1.