Expressions for the Moore–Penrose inverse of block matrices involving the Schur complement

We present a formula for the Moore–Penrose inverse of a matrix of the form M=XNYM=XNY, where X and Y   are nonsingular. Based on this result, we develop explicit expressions for the Moore–Penrose inverse of a 2×22×2 complex block matrix M. We recover the cases when a Schur complement in M is either equal to zero or nonsingular. Some applications to banded matrices and bordered matrices are indicated.