Body On The Two Types Of 2 ¡Á 2 Anti-triangular Block Matrix Group Inverse

In the past few decades, the theory of the generalized inverse of block matrices has been a substantial growth. It has applied in many different areas such as singular differential, difference equations, Markov chains, iterative methods, control theory and cryptography, etc. But the work is so difficult that there are many problems hasn't been solved completely even in the generalized inverse theory. Especially, the existence and representation for the group inverse and Drazin inverse of the block matrix are still very active research topics.Suppose K is a skew field, if A E Knxn, the matrix X E Knxn satisfying AX A=A, XAX=X, AX=XA is called the group inverse of A and is denoted by A#. It is easy to know that if A#exists then A#is unique. For2x2block matrix using the decomposition of matrices, people has researched the existence and representation for the group inverse of this matrix and got many very useful conclusions.In the foundation of original theory, we give the existence and explicit repre-sentations for the group inverse of two following classes2x2anti-triangular block matrices which generalized some related results. exists and R(A)=R(B); and N(B)=N(C)(i) rank(S)=rank(BÏ€A);(ii) rank(S)=rank(ABÏ€);(iii) rank(S)=rank(BÏ€A)=rank(ABÏ€).