Proving that a matrix product is singular

Question

I just played around in mathematica and found out that it seems like if $A$ is an $m\times n$ matrix and B is an $n\times m$ matrix, with $m>n$, then $AB$ is singular. How does one go about proving this?

Answer 1

Looking at these matrices as maps, we have that $\;B:\Bbb F^m\to\Bbb F^n\;$ , and thus $\;\dim\text{Im}\,B\le n\;$ , so $\;\dim\text{Im}\,A(B)\;\le n$ .

Or shortly: linear maps (or matrices) cannot increase the dimension of the domain.