## Nonsingularity of the difference and the sum of two idempotent matrices.(English)Zbl 1195.15005

It is shown that for any two idempotent matrices $$P, Q$$ the nonsingularity of $$P-Q$$ is equivalent to the nonsingularity of any combination $$aP+bQ-cPQ$$ ($$a, b\neq 0, \;a+b=c$$), and the nonsingularity of $$P+Q$$ is equivalent to the nonsingularity of any combination $$aP+bQ-cPQ$$ ($$a, b\neq 0,\;a+b\neq c$$).

### MSC:

 15A09 Theory of matrix inversion and generalized inverses 15A03 Vector spaces, linear dependence, rank, lineability 15A24 Matrix equations and identities

### Keywords:

idempotent matrix; direct sum; rank of matrix
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### References:

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