Idempotency of linear combinations of three idempotent matrices, two of which are commuting

The considerations of the present paper were inspired by Baksalary [O.M. Baksalary, Idempotency of linear combinations of three idempotent matrices, two of which are disjoint, Linear Algebra Appl. 388 (2004) 67–78] who characterized all situations in which a linear combination P=c1P1+c2P2+c3P3, with ci, i=1,2,3, being nonzero complex scalars and Pi, i=1,2,3, being nonzero complex idempotent matrices such that two of them, P1 and P2 say, are disjoint, i.e., satisfy condition P1P2=0=P2P1, is an idempotent matrix. In the present paper, by utilizing different formalism than the one used by Baksalary, the results given in the above mentioned paper are generalized by weakening the assumption expressing the disjointness of P1 and P2 to the commutativity condition P1P2=P2P1.