The Group Inverse Of The Linear Combinations Of Idempotent Matrices

In this note, we mainly discuss explicit expressions for the group inverse of the linear combinations of idempotent matrices under some conditions. Also, we investigate all possible situations when the linear combinations of idempotent matrices is group involutory. The paper is organized as follows:At first, we discuss the expressions for the group inverse of the linear combination aP+bQ+cPQ+dQP+ePQP+fQPQ+gPQPQ+hQPQP of the idempotent matrices P and Q under the conditions (PQ)2= (QP)2, (PQ)2= 0 and so on, where a, b, c, d, e, f,g,h∈C and a≠0,b≠0. And we present some necessary and sufficient conditions for the existence of the group inverse of aP+bQ+cPQ.Also, we discuss all possible situations about the group involutory matrix for the linear combinations aP+bQ+cPQ+dQP+ePQP+fQPQ+gPQPQ+hQPQP of idempotent matrices P and Q under the conditions(PQ)2=(QP)2, (PQ)2=0 and (QP)2≠0, (PQ)2=≠0 and (QP)2=0, where a, b, c, d, e,f, g, h∈C and a≠0, b≠0.