On linear combinations of two tripotent, idempotent, and involutive matrices

Let A=c1A1+c2A2, where c1, c2 are nonzero complex numbers and (A1,A2) is a pair of two n×n nonzero matrices. We consider the problem of characterizing all situations where a linear combination of the form A=c1A1+c2A2 is (i) a tripotent or an involutive matrix when A1andA2 are commuting involutive or commuting tripotent matrices, respectively, (ii) an idempotent matrix when A1andA2 are involutive matrices, and (iii) an involutive matrix when A1andA2 are involutive or idempotent matrices.