## The group inverse of the combinations of two idempotent matrices.(English)Zbl 1220.15008

The authors analyze the group inverse of $aP+bQ+cPQ+dQP+ePQP+fQPQ+gPQPQ$ of two different idempotent matrices $$P$$ and $$Q$$, where $$a, b, c, d, e, f, g \in \mathbb{C}$$ and $$a \neq 0$$, $$b \neq 0$$. They derive its explicit expressions in two cases: $$a+b+c+d+e+f+g \neq 0$$ and $$a+b+c+d+e+f+g = 0$$, under the conditions $$(PQ)^2=(QP)^2$$ and $$(PQ)^2=0$$ or $$(QP)^2=0$$.
The authors also obtain some necessary and sufficient conditions for the existence of the group inverse of $$aP+bQ+cPQ$$ and discuss its explicit expressions.

### MSC:

 15A09 Theory of matrix inversion and generalized inverses 15A27 Commutativity of matrices

### Keywords:

group inverse; idempotent matrix; linear combination
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### References:

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