TITLE: Prove rank(AP) = rank(A) if P is an invertible n × n matrix and A is any m × n matrix?
QUESTION [6 upvotes]: I know how to prove 

But what about we have different size of AP matrix?

REPLY [2 votes]: We can use the result $\mathrm{rank}(AB)\leq\mathrm{min}\{\mathrm{rank}(A),\,\mathrm{rank}(B)\}$ as follows:
$\mathrm{rank}(A) = \mathrm{rank}(A\underbrace{PP^{-1}}_{=I}) \leq \mathrm{rank}(AP) \leq \mathrm{rank}(A).$
Thus $\mathrm{rank}(AP) = \mathrm{rank}(A).$