An element a in ring R has gs-Drazin inverse if there exists b ? comm2(a) such that = b2a, ab Rqnil. For any s C(R), we completely determine when generalized matrix A Ks(R) over local inverse.

In this paper, a review of development the Drazin inverse for sum two matrices has been given. Since topic is closely related to problem finding 2x2 block matrix, paper also offers survey subject.

In this paper, we develop conditions under which the Sherman–Morrison–Woodbury formula can be represented in the Moore–Penrose inverse and the generalized Drazin inverse forms. These results generalize the original Sherman–Morrison–Woodbury formula. © 2011 Elsevier Ltd. All rights reserved.