Any Ring Types 2 × 2 Block Matrix Group Inverse

The research of generalized inverse matrix is very active in the matrix theory field. And it is widely applied in the research of numerical analysis, mathematical statistics, surveying and optimization etc.This paper mainly studies the guoup inverse for two classes of2x2block matrices over arbitrary ring. The first class is in which A#exists,AX=XA and X is invertible.The second class is a2x2block matrix which is with a invertible block and includes the following situations:(1)A is invertible,S#exists,where S=D-CA-1B;(2)D is invertible,S#exists, where S=A-BD⊥C;(3)B or C is reversible.This paper mainly discusses the existence of group inverse of the above two classes of block matrices under various conditions, and gives the necessary and suffi-cient condition. At the same time if the group inverse exists, the paper will give the explicit expression of the group inverse. These results will generalize some results published in recent years.