Some results on the group inverse of the block matrix with a sub-block of linear combination or product combination of matrices over skew fields
Changjiang Bu, Kuize Zhang, Jiemei Zhao · Linear and Multilinear Algebra · 2010
Let K be any skew field and K m×n be the set of all the m × n matrices over K. In this article, necessary and sufficient conditions are given for the existence of the group inverses of block matrix , where and exist, A = c 1 B + c 2 C, non-zero elements c 1 and c 2 are in the centre of K and block matrix , where A = B k C l , k and l are positive integers. Then the representations of the group inverses of these block matrices are also given.