Solve for $X$ in matrix equation

Question

How can I solve for $X$ in this matrix equation?

$$\begin{bmatrix}-3&-8\\-9&5\end{bmatrix} X + \begin{bmatrix}4&-7\\3&-2\end{bmatrix} = \begin{bmatrix}5&8\\-1&-1\end{bmatrix} X$$

First I tried $AX + B = CX$ but then I don't know how to solve for $X$ because no matter what I do I end up getting rid of the $X$.

Answer 1

Hint: You were on the right track. It's quite analogous to solving linear equations with real variables.

$$AX+B = CX$$ $$CX - AX = B$$ $$(C-A)X = B$$ $$X = (C-A)^{-1}B$$ provided that $C-A$ is invertible.

Answer 2

Write the matrix equation in the form

$$A X = B$$

where $X, A, B \in \mathbb{R}^{2 \times 2}$. Vectorizing, one obtains a linear system of $4$ equations in $4$ unknowns

$$(I_2 \otimes A) \, \mbox{vec} (X) = \mbox{vec} (B)$$

Lastly, use Gaussian elimination.

Answer 3

Did you consider, since these are 2 by 2 matrices, just writing X as $\begin{bmatrix}a & b \\ c & d \end{bmatrix}$ and then doing the indicated operations?

$\begin{bmatrix}-3 & -8 \\ -9 & 5\end{bmatrix}$$\begin{bmatrix}a & b \\ c & d \end{bmatrix}+ \begin{bmatrix}4 & -7 \\ 3 & -2 \end{bmatrix}$$= \begin{bmatrix}5 & 8 \\ -1 & -1\end{bmatrix}\begin{bmatrix}a & b \\ c & d \end{bmatrix}$.

$\begin{bmatrix}-3a- 8c+ 4 & -3b- 8d- 7 \\ -9a+ 5b+ 3 & -9b+ 5d- 2\end{bmatrix}= \begin{bmatrix}5a+ 8c & 5b+ 8d \\ -a- c & -b- d\end{bmatrix}$.

So we have the system of four equations

$$-3a- 8c+ 4= 5a+ 8c\\-9a+ 5b+ 3= -a- c\\-3b- 8d- 7= 5b+ 8d\\-9b+ 5d- 2= -b- d$$

Which is easy to solve for $a, b, c,$ and $d$.