Given two matrices $A_{m\times n}$ and $B_{n\times p}$, what is the sufficient and necessary condition for $AB$ to have full rank?
I know $r(AB)=r(B)-\dim N(A) \cap R(B)$, so is it true the above iff $\dim N(A)\cap R(B)=0$? But seems incorrect if $m\lt p$. Please help.