Solving the equation $ XA = A + 2X $

Question

How to solve $XA = A + 2X$ if $A$ being: \begin{equation} A= \begin{bmatrix} 1 & 0 & 1 \\ 0 & 2 & 1 \\ 0 & 1 & 1 \\ \end{bmatrix} \\ \end{equation}

I tried going $X = (A+2X)A^{-1}$ but I don't know how to get the 2X out from there.

Answer 1

So you have $X(A-2I)=A$ thus $$ X = A\cdot (A-2I)^{-1}=...$$

Notice that $ (A-2I)^{-1}$ exists since $$A-2I = \begin{bmatrix} -1 & 0 & 1 \\ 0 & 0 & 1 \\ 0 & 1 & -1 \\ \end{bmatrix} \implies \det(A-2I)= 1$$