Show that $rank(A)+rank(B) \leq n$, when $A,B$ are $2$ matrices of size $n \times n$, and $AB=0$

Question

Question from homework in Linear Algebra:

Let $A,B$ be two matrices of size $n \times n$ such that $AB=0$.

Show that: $rank(A) + rank(B) \le n$ .

It probably has something to do with the dim of the null space or column space but I can't put things together from what we've learned...

Please help.. Thanks. :)

Answer 1

Hint: show that $\operatorname{Im}(B)\subset \ker A$ and a well-known formula linking the rank and the dimension of the kernel of a matrix with the dimension of the underlying space.