Group inverses for some 2 × 2 block matrices over rings

Abstract

We first consider the group inverses of the block matrices \left(\begin{array}{l}A\\ \mathrm{0}\end{array}\begin{array}{l}B\\ C\end{array}\right) over a weakly finite ring. Then we study the sufficient and necessary conditions for the existence and the representations of the group inverses of the block matrices \left(\begin{array}{l}A\\ C\end{array}\begin{array}{l}B\\ D\end{array}\right) over a ring with unity 1 under the following conditions respectively: (i) B = C, D = 0, B^# and (B^πA) ^# both exist; (ii) B is invertible and m = n; (iii) A^# and (D - CA^#B)^# both exist, C = CAA^# , where A and D are m × m and n × n matrices, respectively.

Keywords

Group inverse / block matrix / right Ore domain / associative ring

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