Let $A,B$ be some real matrices, each $n\times n$. Given that $$rank(A) + rank(B) \le n,$$ show that there exists a real $n \times n$ matrix $C$, with $rank(C) = n$, such that $ACB = 0$.
I cannot figure out how to show that such a matrix exists. So far I have tried solving the problem using Forbenius inequality but I cannot prove that such a matrix exists. How I can approach such a problem?