Let E, F ? Cnxn. If EFiE = 0 for all i N, we give the explicit representation of Drazin inverse block complex matrix (E I 0). We thereby solve a wider kind singular differential equations posed by Campbell [S.L. Campbell, The and systems second order linear equations, Linear & Multilinear Algebra, 14(1983), 195-198].

We study properties of the Drazin index of regular elements in a ring with a unity 1. We give expressions for generalized inverses of 1−ba in terms of generalized inverses of 1−ab. In our development we prove that the Drazin index of 1 − ba is equal to the Drazin index of 1 − ab. 2000 Mathematics subject classification. primary 15A09; secondary 16U99.