Let $A$ and $B$ be given $(n\times m)$-matrices ($n\geq 2$, $m\geq 2$), and let $X$ be an unknown $(n\times n)$-matrix such that $$XA=B.$$ When does such a solution $X$ of this matrix equation exist? When does $X$ exist and is unique?
As I see, one has $$A^T X^T=B^T.$$ So, the necessary and sufficient condition for existence of $X$ is $$\mathrm{rank}(A^T)=\mathrm{rank}(A^T|B^T)$$ or (that is the same) $$\mathrm{rank}(A)=\mathrm{rank} \Bigg(\frac{\boxed{A}}{\boxed{B}}\Bigg).$$ Is it possible to simplify this condition or write simpler conditions?