Tensor product of matrix

Question

Let $ F$ a field, $A\in M_n(F) $ and $B\in M_m(F) $. Two matrix over $F$.

We have $ A\otimes B \in M_{nm}(F) $ tensor product of $A$ and $B$

Question:

For any matrix $C \in M_4(F) $ can I find two matrix $A, B \in M_2(F)$ such that

$A\otimes B = C$

Answer 1

In general no.

For instance, the rank of $A\otimes B$ is the rank of $A$ multiplied by the rank of $B$. So a matrix $C$ of rank $3$ in $M_4(F)$ cannot possibly be an example of $A\otimes B$.