Suppose two $n\times n$ matricies, $A$ and $B$, how many possible solutions are there. Suppose i construct a $n\times n$ matrix $C$, by multiplying two $n\times n$ matrices $A$ and $B$ i.e. $AB = C$. Given $B$ and $C$, how many other $A$'s can yield $C$ also i.e. is it just the exact $A$, infinitely many other $A$'s or no other $A$'s. There are no assumptions made about the invertability of $A$ and $B$. In the case that $A$ and $B$ are invertable there exists only one such $A$. 

If $B$ is not invertible its range is a proper subspace; if $a$ is any matrix which maps the range of $B$ to $\{0\}$ then $(A+a) \cdot B = A \cdot B$. There are infinitely many such $a$. On the other hand if $D \cdot B = A \cdot B$ then $(D-A) \cdot B = 0$, i.e. $D-A$ maps the range of $B$ to $\{0\}$.
