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?
What can you say about the rank of $B$ relative to the nullity of $A$?
Note that if $U$ and $W$ are any two subspaces of the same dimension, there's a full-rank map on $n$-space that sends $U$ to $W$. Proof: Pick a basis $u_1, \ldots, u_k$ of $U$; extend it by $ a_1, \ldots, a_{n+k}$ to a basis of $n$-space. Do the same for $W$. Now send $u_i$ to $w_i$ and $a_i$ to $b_i$ for each $i$ that makes sense, and you've defined the isomorphism.