Can rank(AB) < rank(A) and rank(AB) < rank(B) hold for matrices A and B?

Question

I think that the statement is true because every row in AB is a linear combination of B and every column in AB is a linear combination of A. Therefore C(AB) ⊆ C(A) and R(AB) ⊆ R(B) so rank(AB) must be <= min(rank(A), rank(B)) Although I'm not sure whether my proof is enough and right. How else would you prove it?

Thank you very much for help!

Answer 1

Oh yes! Just take any non-zero matrix $A$ such that $A^2=0$, for instance $$A=\begin{pmatrix} 0&1\\0&0\end{pmatrix}.$$