Solve a matrix equation

Question

I need to find $X$ from

$$\begin{pmatrix} 1 & 2\\ -3 &-6 \end{pmatrix} X \begin{pmatrix} 1 &2 \\ -1 &-2 \end{pmatrix}=\begin{pmatrix} 2 &4 \\ -6 & -12 \end{pmatrix}$$

I wrote $X$ as

$$X=\begin{pmatrix} a & b\\ c &d \end{pmatrix}$$

and I got $a+2c-b-2d=2$ but I do not know what to do next. Please help.

Answer 1

You are on good way. When you multiply matrices on left side, you finally get $$a-b+2c-2d = 2$$ which is a linear equation. Solutions are given with three parameters, and thus

$$ X = \begin{pmatrix} a & b\\ c & \frac{a-b+2c-2}{2} \end{pmatrix}$$