Drazin Inverse Of Some 2 ¡Á 2 Block Matrix Ring

Since the Drazin inverse of square matrix always exists on a complex numberfield and skew field, we mainly research the representation for the Drazin inverse.Because the Drazin inverse of matrix may not exist, it is worthwhile to study theconditions of the existence for the Drazin inverse of matrix over general Algebrastructure, we mainly discuss the Drazin inverse and the group inverse of some blockmatrices over an arbitrary ring.Let A,X are square matrices over rings, and k be a positive integer, if A andX satisfy :(?)then X is called the k-Drazin of A, denoted by ADk. It is well known if ADk exists,it is unique. When k = 1, then X is called the group inverse of A, denoted by A#.This paper is divided into three sections, in the first section, we introduce therelative definitions of the Drazin inverse of matrix and its wider applications, andoutline the study conclusions of the Drazin inverse of matrix over rings and thetrends of development, thus clearly determine the importance for the Drazin inverseof matrix over rings or more general rings. In the second section, we researchthe index and the Drazin inverse of 2@2 partitioned matrix over a skew field andgeneralize some results. In the third section, we study the conditions of the existenceand representation for the Drazin inverse and the group inverse of the following threeblock matrices over an associative ring with 1.