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A,B matrix nxn so that AB=0 proof that rank(A)+rank(B)=n I don't know even where to start, or look for.

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What if $A$ and $B$ are both the zero matrix? Are you sure of the statement? –  Dylan Moreland Jan 10 '12 at 17:39

1 Answer 1

Let's assume the problem is the following:

Given two $(n\times n)$-matrices $A$, $B$ with $AB=0$, prove that ${\rm rank}(A) +{\rm rank}(B)\leq n$.

A hint: $AB=0$ means the following: Any vector in ${\rm im}(B)$ is annihilated by $A$, in other words: ${\rm ker}(A)\supset{\rm im}(B)$. What does this imply for the dimensions of the involved spaces and the ranks of the involved maps/matrices ?

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