Published online by Cambridge University Press: 08 November 2013
Let ${ \mathbb{K} }^{m\times n} $ denote the set of all $m\times n$ matrices over a skew field $ \mathbb{K} $. In this paper, we give a necessary and sufficient condition for the existence of the group inverse of $P+ Q$ and its representation under the condition $PQ= 0$, where $P, Q\in { \mathbb{K} }^{n\times n} $. In addition, in view of the natural characters of block matrices, we give the existence and representation for the group inverse of $P+ Q$ and $P+ Q+ R$ under some conditions, where $P, Q, R\in { \mathbb{K} }^{n\times n} $.