Idempotency of linear combinations of an idempotent matrix and a t-potent matrix that commute

This paper deals with idempotent matrices (i.e., A2 = A) and t-potent matrices (i.e., Bt = B). When both matrices commute, we derive a list of all complex numbers c1 and c2 such that c1A + c2B is an idempotent matrix. In addition, the real case is also analyzed.