Nonsingularity of the difference and the sum of two idempotent matrices

Groß and Trenkler 1, pointed out that if a difference of idempotent matrices P and Q is nonsingular, then so is their sum, and Koliha et al 2 expressed explicitly a condition, which combined with the nonsingularity of P+Q ensures the nonsingularity of P-Q. In the present paper, these results are strengthened by showing that the nonsingularity of P-Q is in fact equivalent to the nonsingularity of any combination aP+bQ-cPQ (where a≠0,b≠0,a+b=c), and the nonsingularity of P+Q is equivalent to the nonsingularity of any combination aP+bQ-cPQ (where a≠0,b≠0,a+b≠c).