Consider the system of equations $$ Xy=Ab $$ where $X$ and $A$ are $m \times m$ invertible matrices and $y$ and $b$ are $m \times n$ matrices. The matrices $X$ and $y$ are unknown and the matrices $A$ and $b$ are given. Under what restrictions on $X$ (with the same restrictions on $A$) and / or $y$ (with the same restrictions on $b$) do $X=A$ and $y=b$? For example, if we restrict $X$ (and $A$) to be the identity matrix then $X=A$ and $y=b$. What if we impose that $X$ and $A$ must be some band matrices with ones on the diagonal or some sparse matrix?
A necessary condition is the number of unknowns is less than or equal to $m \times n$ because $Ab$ provides $m \times n$ equations.