Chapter 4, Problem 39RE

One solution to the input-output equation $X=MX+D$ is given by $X={\left(I-M\right)}^{-1}D$ Discuss the validity of each step in the following solutions of this equation. (Assume that all necessary inverses exist.) Are both solutions correct?

$\begin{array}{c}\left(\text{A}\right)X=MX+D\\ X-MX=D\\ X\left(I-M\right)=D\\ X=D{\left(I-M\right)}^{-1}\end{array}$

$\begin{array}{c}\left(\text{B}\right)X=MX+D\\ -D=MX-X\\ -D=\left(M-I\right)X\\ X={\left(M-I\right)}^{-1}\left(-D\right)\end{array}$