TITLE: Yet another matrix equation
QUESTION [6 upvotes]: Solve the following matrix equation for $X$.
$$\left[\begin{array}{cc}
5 &-8\cr
8 &1
\end{array}\right] X + \left[\begin{array}{cc}
6 &6\cr
3 &5
\end{array}\right] = \left[\begin{array}{cc}
-1 &4\cr
-3 &-1
\end{array}\right] X$$

Please give me some hint to do this question. Thanks.

REPLY [18 votes]: Your matrix equation is $$ AX+B=CX$$
You solve for X.
$$(A-C)X=-B$$
$$X=-(A-C)^{-1}B$$
$$X=
\left[\begin{array}{cc}
-6 &12\cr
-11 &-2
\end{array}\right]^{-1}\left[\begin{array}{cc}
6 &6\cr
3 &5
\end{array}\right]$$
$$ X = \frac{1}{12}\left[\begin{array}{cc}
-4 & -6\cr
4 &3
\end{array}\right]$$