On some linear combinations of commuting involutive and idempotent matrices

We consider the problem of characterizing situations where a linear combination of commuting involutive matrices is a k-potent matrix, where k   is a positive integer such that k⩾2k⩾2. Also, we consider the problem of characterizing situations where a linear combination of the form c1In+c2A+c3Bc1In+c2A+c3B is a nonsingular or an involutive matrix, when idempotent matrices A,B∈Cn×n satisfy different conditions and c1,c2,c3∈C⧹{0}.