$ABAB=0_{n\times n}$, 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"

My idea for a counterexample: For $n=3$: take $A$ to be the homomorphism that kills the first line of a vector, $B$ the homomorphism that moves every line one up, but the first line which will stay in place. This can be generalized for $n \geq 3$.
Edit: As I was typing this question was still without answer. Of course the answer above is just the matrix version of what I wrote (with the slight difference that above $B$ actually 'kills' the first line)
