The nullity and rank of linear combinations of idempotent matrices

Baksalary and Baksalary [J.K. Baksalary, O.M. Baksalary, Nonsingularity of linear combinations of idempotent matrices, Linear Algebra Appl. 388 (2004) 25–29] proved that the nonsingularity of P1 + P2, where P1 and P2 are idempotent matrices, is equivalent to the nonsingularity of any linear combinations c1P1 + c2P2, where c1, c2 ≠ 0 and c1 + c2 ≠ 0. In the present note this result is strengthened by showing that the nullity and rank of c1P1 + c2P2 are constant. Furthermore, a simple proof of the rank formula of Groß and Trenkler [J. Groß, G. Trenkler, Nonsingularity of the difference of two oblique projectors, SIAM J. Matrix Anal. Appl. 21 (1999) 390–395] is obtained.