Rank of a matrix based on invertible

Question

How does $\operatorname{rank} [AB]$ depend on $\operatorname{rank}[A]$ & $\operatorname{rank}[B]$ when

$A$ is invertible; $A$ & $B$ both are invertible; $A$ & $B$ both are non-invertible

Answer 1

It always holds that $$\DeclareMathOperator{\rank}{rank} \rank(AB)\le\min\{\rank(A),\rank(B)\}\tag{*} $$ but nothing more can be said in general.

If $A$ is invertible, then we can say $$ \rank(B)=\rank(A^{-1}(AB))\le\rank(AB)\le\rank(B) $$ because of (*), so $\rank(AB)=\rank(B)$.

If $A$ and $B$ are invertible $n\times n$ matrices, then they both have rank $n$.