Chapter 4.7, Problem 18E

Problems $15\text{-}20$ pertain to the following input-output model: Assume that an economy is based on three industrial sectors: agriculture $\left(A\right)$, building $\left(B\right)$. and energy $\left(E\right)$.The technology matrix $M$ and final demand matrices (in billions of dollars) are

$\begin{array}{l}\begin{array}{cccc}& A& B& E\end{array}\\ \begin{array}{c}A\\ B\\ E\end{array}\left[\begin{array}{ccc}0.3& 0.2& 0.2\\ 0.1& 0.1& 0.1\\ 0.2& 0.1& 0.1\end{array}\right]=M\end{array}$

$D_1=\left[\begin{array}{c}5\\ \begin{array}{l}10\\ 15\end{array}\end{array}\right]D_2=\left[\begin{array}{c}20\\ \begin{array}{l}15\\ 10\end{array}\end{array}\right]$

Find ${\left(I-M\right)}^{-1}.$

Show that ${\left(I-M\right)}^{-1}\left(I-M\right)=I.$