Values of rank $AB$

Question

If $A$ is a $4\times2$ matrix and $B$ is a $2\times 3$ matrix, what are the possible values of $\operatorname*{rank}(AB)$?

Construct examples of $A$ and $B$ exhibiting each possible value of $\operatorname*{rank}(AB)$ and explain your reasoning.

Answer 1

Hint. Use that in general, $$\text{rank}(AB)\leq \min\{\text{rank }A, \text{rank }B\},$$ and in our case, $$0\le \text{rank }A \le 2,\quad 0\le \text{rank }B \le 2.$$

Answer 2

rank = number of pivots in reduced form

Okay, so in terms of pivots per column, matrix A can have at most 2 pivots and matrix B also 2 pivots at most. AB which is 4*3 can have 3 pivots at most so I don't really understand why the formula above is true.