The following is a problem from Daniel Norman's Introduction to Linear Algebra. Consider a $m\times n$ matrix, $M$, with rank $r$. We wish to show that the matrix $M^\mathrm{T}M$ also has rank $r$. Now, I can show this fact by proving that the nullspace of $M^\mathrm{T}M$ is the same as the nullspace of $M$ whereby the rank-nullity theorem will take care of the rest, but the hint of the problem proposes something confusing to me.
The hint suggests looking at the matrix $N$ obtained by exchanging two columns of $M$. It then asks "What is the relationship between $N^\mathrm{T}N$ and $M^\mathrm{T}M$?"
I can see that the two matrices are orthogonally similar, more specifically I can see that
$E^\mathrm{T}M^\mathrm{T}ME = N^\mathrm{T}N$ where $E = E^\mathrm{T}$ is an elementary matrix exchanging the columns in question. But I fail to see anything which leads to a resolution of this problem. Can anyone offer some help?