Suppose the complex matrix M is partitioned into a $2 \times 2$ array of blocks; let $M_{11} = A,M_{12} = B,M_{21} = C,M_{22} = D$. The generalized Schur complement of A in M is defined to be $M/A = D - CA^ + B$, where $A^ + $ is the Moore–Penrose inverse of A. The relationship of the ranks of M, A, and $M/A$ is determined. A new proof, under certain conditions, of Sylvester’s determinantal formula is given. A quotient formula like one previously proved for the Schur complement is obtained. Finally, several known inequalities for positive semidefinite Hermitian matrices are generalized.