$QR$ factorization: why $A$ and $R$ have the same rank

Question

In $A=QR$ factorization why $R$ has always the same rank of $A$?

Answer 1

this is because $rank(AB) \le min\{rank(A), rank(B)\}$. This gives you an inequality one way, and you get the other way by inverting $Q$ and looking at $R = Q^{-1}A$