Hint: Think about determinants (det(BA) = det(AB)).
I thought so, but now I realize I'm confused and I think it's false. Disregard my post.
EDIT: If rank (AB) is one, there are 2 null rows in the equivalent reduced matrix of AB, right? I have a conclusion but I don't know if it's true. I'll wait someone to help us.
EDIT 2 : we were on the right track. det (AB)=0 since the its rank is 1. Hence det (BA) =0 and so it has at least a null row. What does it say about the rank?