Find matrix such that:

Question

Find matrix such that: $$\begin{pmatrix} 3 & 2 & 3 \\ 3 & 6 & 3 \\ 1 & 2 & 4 \end{pmatrix} X+ \begin{pmatrix} 1 & 1 & 0 \\ 1 & -1 & 2 \\ 1 & 0 & 1 \end{pmatrix}=2X $$ inverse of which matrix i have to find? How to approach that?

Answer 1

Hint:

Notice that $$ AX +B = 2X \implies (A-2I)X = -B \implies X = - (A-2I)^{-1}B.$$

Answer 2

Hint: you should write it like that:

$\begin{pmatrix} 3 & 2 & 3 \\ 3 & 6 & 3 \\ 1 & 2 & 4 \end{pmatrix} X+ \begin{pmatrix} 1 & 1 & 0 \\ 1 & -1 & 2 \\ 1 & 0 & 1 \end{pmatrix}=\begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \\ \end{pmatrix}X$

or

$\begin{pmatrix} 1 & 2 & 3 \\ 3 & 4 & 3 \\ 1 & 2 & 2 \end{pmatrix} X=\begin{pmatrix} -1 & -1 & 0 \\ -1 & 1 & -2 \\ -1 & 0 & -1 \end{pmatrix}$