the product of a matrix and a permutation matrix

Question

Can a permutation matrix ($P$) be used to change the rank of another matrix ($M$)? Is there any literature to this effect, or to the contrary?

I've tried a few small examples and the resulting matrix ($M_2$) seems to always have the same rank as the input matrix ($M$)

$M_2 = M P$

Answer 1

Hint: The rank of a matrix is the number of linearly independent row vectors, or of linearly independent column vectors. Now think about what a permutation matrix does to the row or column vectors in the matrix if you multiply it from left or right.

Answer 2

Hint If $A\in M_n(\mathbb R)$ and $B\in GL_n(\mathbb R)$ then $$\mathrm{rank} (AB)=\mathrm{rank} (A)=\mathrm{rank} (BA)$$