Q: nullity of matrices If A ,is an n x m matrix and B is an m x n matrix, then nullity(AB) =
nullity(BA).
Determine if the statement is true.
For this question, how do I go about finding the solution? What's the thought process? I never get the type of questions where the required answer is to prove a statement right or wrong, it feels like plain guess and check.

A: Let $n=m=2$ and
$$A:=\begin{pmatrix}0& 0\\
   0 & 1\\ \end{pmatrix} \ \ \text{and} \ \
B:=\begin{pmatrix}0&1\\
   0 & 0\end{pmatrix}.\tag{2}$$
Then we have $AB=0$ but $BA \ne 0$.This shows that
$$nullity(AB)  \ne  nullity(BA).$$
