A representation for the Drazin inverse of block matrices with a singular generalized Schur complement

Consider a 2×22×2 block complex square matrix M=ABCD, where A and D   are square matrices. Suppose that (I-AAD)B=O(I-AAD)B=O and C(I-AAD)=OC(I-AAD)=O, where ADAD is the Drazin inverse of A  . The representations of the Drazin inverse MDMD have been studied in the case where the generalized Schur complement, S=A-CADBS=A-CADB, is either zero or nonsingular. In this paper, we develop a representation, under certain conditions, for MDMD when S   is singular and group invertible. Moreover, this formula includes the case where S=OS=O or nonsingular. A numerical example is given to illustrate the result.