### Alternative characterizations of some linear combinations of an idempotent matrix and a tripotent matrix that commute

Tuğba Petik, Burak Tufan Gökmen

#### Öz

In this work, first, Theorem 2 in [1] [Yao, H., Sun, Y., Xu, C., and Bu, C., A note on linear combinations of an idempotent matrix and a tripotent matrix, J. Appl. Math.  Informatics, 27 (5-6), 1493-1499, 2009] and Theorem 2.2 in [2][Özdemir H., Sarduvan M., Özban A.Y., Güler N., On idempotency and tripotency of linear combinations of two commuting tripotent matrices, Appl. Math. Comput., 207 (1), 197-201, 2009] are reconsidered in different ways under the condition that the matrices involved in the linear combination are commutative. Thus, it is seen that there are some missing results in Theorem 2 in [1]. Then, by considering the obtained results and doing some detailed investigations, it is given a new characterization, without any restriction on the involved matrices except for commutativity, of a linear combination of an idempotent and a tripotent matrix that commute.

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#### Referanslar

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