Chapter 4.7, Problem 18E

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

$\begin{array}{l}ABE\\ \begin{array}{c}A\\ B\\ E\end{array}\left[\begin{array}{l}0.30.20.2\\ 0.10.10.1\\ 0.20.10.1\end{array}\right]=M\\ D_1=\left[\begin{array}{l}5\\ 10\\ 15\end{array}\right]D_2=\left[\begin{array}{l}20\\ 15\\ 10\end{array}\right]\end{array}$

Given

${\left(I-M\right)}^{-1}=\left[\begin{array}{l}1.60.40.4\\ 0.22\text{ 1}\text{.18 }0.18\\ 0.380.22\text{ 1}\text{.22}\end{array}\right]$

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