There is a remark my professor made in his notes that I simply can't wrap my head around. He even gave a proof but it made me even more confused. Here it is:

Two matrices:

Q is invertible

AQ is mxn

rank(AQ) = rank A

Note:I'm confused by this alone (I'll show the proof after if someone can help me understand this first). What is the relevance of Q's invertibility?