Representations of generalized inverses of partitioned matrix involving Schur complement

In this article, we consider some representations of {1,3},{1,4},{1,2,3}{1,3},{1,4},{1,2,3} and {1,2,4}{1,2,4}-inverses of the partitioned matrix M which are equivalent to some rank additivity conditions. We present the application of these results to generalizations of the Sherman–Morrison–Woodbury-type formulae.