Matrices & Systems · Challenge problem

In a two-sector economy, producing one unit of goods uses up $\tfrac15$ unit of goods and $\tfrac{3}{10}$ unit of services, and producing one unit of services uses up $\tfrac25$ unit of goods and $\tfrac1{10}$ unit of services. These inputs form the consumption matrix $C = \begin{pmatrix}1/5 & 2/5\\3/10 & 1/10\end{pmatrix}$ (column $j$ lists the inputs for one unit of sector $j$). Find the sum of all four entries of the matrix series $I + C + C^2 + C^3 + \cdots$.