# Rank of a matrix $A$ such that $A + A^T = 0$

I need to prove (using only elementary operations and induction) that rank of a matrix $A\in \Bbb C^{n\times n}$ such that $A + A^T = 0$ is an even number. I know that elementary operations doesn't change a rank but I don't know how to use them in such a problem. Any hints?

• I need to prove it using only elementary operations and induction. Unfortunately, the answer in that topic uses subject which hasn't been covered on my course. – wisniak Nov 22 '14 at 13:00
• @wisniak You should probably edit the question then. – Algebraic Pavel Nov 22 '14 at 13:08
• What's the ground field? Is it $\mathbb R$? Does it have characteristic $2$? – jflipp Nov 22 '14 at 13:23