A note on the representations for the Drazin inverse of 2 × 2 block matrices

In 1979, Campbell and Meyer proposed the problem of finding a formula for the Drazin inverse of a 2 × 2 matrix in terms of its various blocks, where the blocks A and D are required to be square matrices. Special cases of the problems have been studied. In particular, a representation of the Drazin inverse of M, denoted by MD, has recently been obtained under the assumptions that C(I − AAD)B = O and A(I − AAD)B = O together with the condition that the generalized Schur complement D − CADB be either nonsingular or zero. We derive an alternative representation for MD under the same assumptions, but with the condition on the Schur complement in the hypothesis replaced by the condition that R(CAAD)⊂N(B)∩N(D), where R(·) and N(·) are the range and null space of a matrix.