Note: The following questions are from the 5th question of the 2010 Chinese Graduate Mathematical Entrance Examination (first set):
Suppose A is an $m*n$ matrix, B is an $n*m$ matrix, and $AB=E$ (E is the m-order unit matrix), then the following conclusions are correct (r(X) represents the rank of matrix X):
$$ \begin{array}{c} &(A)& r(A)=m, \quad r(B)=m &(B)& r(A)=m, \quad r(B)=n \\ &(C)& r(A)=n, \quad r(B)=m &(D)& r(A)=n, \quad r(B)=n \end{array}$$
At present, I know that $r(E) = m$,$r(A) \le \max(m,n)$.
Clearly, we have $r(A) \leq m$. And we have $m = r(E) = dim(im(E)) \leq dim(im(A)) = r(A)$. Then $r(A) = m$.
We have $B^T A^T = (AB)^T = E^T = E$. Then by the same reasoning, $r(B^T) = m$. Then $r(B) = m$.