If rank(A) = 3 and rank(B) = 3 for two 3x3 matrices, is it possible for rank(AB) to be less than 3?

Question

Matrices A and B are 3 x 3 matrices, and rank(A) = rank(B) = 3. I understand why rank(AB) can be equal to rank(A) and rank(B), but I don't see how rank(AB) could be less.

Answer 1

A $3\times 3$ matrix of rank $3$ is necessarily invertible, and the product of two invertible matrices is invertible and thus of rank $3$.

As for an example of what you're asking for, you can take any nilpotent matrix, for example:

$$N=\begin{pmatrix} 0& 1& 0\\ 0& 0& 1\\ 0& 0& 0 \end{pmatrix}.$$

You see that $\text{rank}(N)=2$ and $\text{rank}(N^2)=1$.