I am learning the MIT ocw 18.06 Linear Algebra, and there is a problem in problem set 5 as below:
Problem 34. If A has r independent columns and B has r independent rows, AB is invertible. Proof: When A is m by r with independent columns, we know that $A^TA$ is invertible. If B is r by n with independent rows, show that $BB^T$ is invertible. (Take A = $B^T$ .) Now show that AB has rank r.
But I think BA is invertible other than AB, so this problem makes me confused. Am I understanding something wrong here? Thanks a lot.