I am having problem in solving set of matrix multiplication. There are three matrices $A,X$ and $Y$, all are non-singular $2\times 2$ matrices. Where matrix $X$ and $Y$ are known and $A$ is unknown. $$ X = \begin{bmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{bmatrix} $$ $$ Y = \begin{bmatrix} y_{11} & y_{12} \\ y_{21} & y_{22} \end{bmatrix} $$ $$ A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} $$
and multiplication is as follows
$$ X\cdot A=A\cdot Y $$
I expanded it and tried to solve it as such I would be able to get elements of matrix $A$ at the end but it end up in homogenous linear equation having trivial solution zero. That makes all calculation meaningless.
$$ X\cdot A - A\cdot Y = 0 $$
What is the better way to compute it as such I can find result of matrix $A$ (in terms of elements of matrics $X$ and $Y$) at the end.