On deriving the Drazin inverse of a modified matrix

This paper addresses the problem of deriving formulas for the Drazin inverse of a modified matrix. First we focus on obtaining formulas of Sherman–Morrison–Woodbury type for singular matrices, dealing with the Drazin inverse. Denote the Drazin inverse of a square complex matrix A by AD. We provide some expressions of (A-CDDB)D in terms of the Drazin inverse of the original matrix A and of its generalized Schur complement Z=D-BADC, where Z is not assumed invertible.In addition, we study some representations of the Drazin inverse of a modified matrix, by using an auxiliary idempotent matrix and some special cases are analyzed. The provided results extend earlier works given in the literature.