$ABAB=0_{n\times n}$, then must $BABA$ be $0_{n\times n}$? 
Let $A$ and $B$ be $n\times n$ matrices such that $ABAB=0_{n\times n}$.  Can we conclude that  $BABA$ must be  $0_{n\times n}$?

I fail to give a counterexample when $n=2$， so I guess the answer is "yes"

For $n = 2$: The characteristic polynomials of $AB$ and $BA$ are the same (for any $n$). Therefore, if $AB$ is nilpotent then $BA$ is also nilpotent. So, we necessarily have $BABA = (BA)^2 = 0$.
