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

In 1979, Campbell and Meyer proposed the problem of finding a formula for the Drazin inverse of a 2 × 2 matrix image 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 image, where image and image are the range and null space of a matrix.