INVOLUTORY AND S+1-POTENCY OF LINEAR COMBINATIONS OF A TRIPOTENT MATRIX AND AN ARBITRARY MATRIX

Let and be nonzero complex matrices, denote a linear combination of the two matrices by , where , are nonzero complex numbers. In this paper, we research the problem of the linear combinations in the general case. We give a sufficient and necessary condition for A is an involutive matrix and s+1-potent matrix, respectively, where is a tripotent matrix, with . Then, using the results, we also give the sufficient and necessary conditions for the involutory of the linear combination A, where is a tripotent matrix, anti-idempotent matrix, and involutive matrix, respectively, and is a tripotent matrix, idempotent matrix, and involutive matrix, respectively, with .