Abstract
In this paper, we study the generalized inverse of a matrix A over an integral domain. We first present a determinantal representation of one {1,5}-inverse of a matrix which has the group inverse. Next, we obtain the necessary and sufficient conditions for the existence of the generalized inverse , an explicit expression for which reduces to one {1,5}-inverse and a determinantal representation of . In addition, we present as a convex combination of solutions of some nonsingular linear equations. Finally, we present two illustrative examples for computing the generalized inverses.
Acknowledgements
The authors would like to thank Professors Fuzhen Zhang and S. Kirkland, and the three referees for their very detailed comments on our previous manuscript. Y. Yu is supported by Science Foundation of Shanghai Municipal Education Commission (CW0519). Y. Wei is supported by National Natural Science Foundation of China under grant 10871051 and Shanghai Science and Technology Committee under grant 08511501703.