Let $\displaystyle A$ and $\displaystyle B$ be two matrices, of order $\displaystyle m$x$\displaystyle n$ and $\displaystyle n$x$\displaystyle m$ respectively, such that $\displaystyle r(AB) = r(BA)= min\{r(A), r(B)\}$. Then $\displaystyle AB$ is idempotent implies $\displaystyle BA$ is also idempotent.

Please help. I am neither able to prove this, nor able to get any counter-example to disprove this.