The rank of skew-symmetric matrix is even

Question

I know that the rank of a skew-symmetric matrix is even. I just need to find a published proof for it. Could anyone direct me to a source that could help me?

Answer 1

You can see Hoffman's book on linear algebra last chapter on Bilinear forms which says rank of skew symmetric matrix is always even.

Answer 2

This is not an answer, but a remark. The rank of a skew symmetric matrix is not always even. It really depends on the ground field. Over a finite field of characteristic 2 (i.e. $GF(2)$; see here for a brief description), we have $1=-1$. Hence the matrix $A=\begin{pmatrix}1&1\\1&1\end{pmatrix}=-\begin{pmatrix}1&1\\1&1\end{pmatrix}$ is skew symmetric. Yet $A=\begin{pmatrix}1\\ 1\end{pmatrix}\begin{pmatrix}1&1\end{pmatrix}$ is also of rank 1.

That said, a complex skew symmetric matrix does have an even rank, as proved in the monographs mentioned by Billy and Dylan Moreland in the comments to your question.