Group invertible block matrices

Abstract

Let M = A B C D (A and D are square) be a 2 × 2 block matrix over a skew field, where A is group invertible. Let S = D − CA # B denote the generalized Schur complement of M. We give the representations and the group invertibility of M under each of the following conditions: (1) S = 0; (2) S is group invertible and CA π B = 0, where A π = I − AA # . And the second result generalizes a result of C.