On idempotency and tripotency of linear combinations of two commuting tripotent matrices

Let T1T1 and T2T2 be two nonzero commuting n × n   tripotent matrices and c1,c2c1,c2 two nonzero complex numbers. Necessary and sufficient conditions for the tripotency and the idempotency of c1T1+c2T2c1T1+c2T2 are obtained. The problems considered here have also statistical importance when c1,c2c1,c2 are real scalars and T1,T2T1,T2 are real symmetric matrices.