I have searched for the above topic and found some results, but the answer I am looking for is not found anywhere. Here is my question:
Given $A_{m \times n}$ matrix with rank $m$, and $B_{n \times p}$ matrix with rank $p$, where $n > p \geq m$. I know that
$$
\operatorname{rank}(AB) \leq
\min\left(\operatorname{rank}(A),\operatorname{rank}(B)\right)
$$
What I want to know is if this expression holds for equality. I.e is this expression
$$
\operatorname{rank}(AB) =
\min\left(\operatorname{rank}(A),\operatorname{rank}(B)\right) = m
$$
correct? If yes, how can it be proved?
Best Answer
if we take A and B be 2 non-zero matrices s.t AB= zero matrix then rank A,rank B >0 but rank of AB = 0 so rank AB is not equal to min{rank A, rank B}