Let $S_1=A-BD^+C$ and $S_2=D-CA^+B$ be the associated Schur complements of $M=\biggl[\smallmatrix A&B\\C&D\endsmallmatrix\bigr]$. We derive necessary and sufficient conditions for $S_1=0$ imply $S_2=0$ by using generalized inverses of matrices and singular value decompositions.