On linear combinations of two tripotent, idempotent, and involutive matrices (Q929459)

Let (1) \(A = c_{1}A_{1} + c_{2}A_{2}\), where \(c_{1}, c_{2}\) are nonzero complex numbers, and (\(A_1, A_2\)) is a pair of two \(n\times n\) nonzero matrices. The purpose of this paper is mainly twofold: in case \(A_1\) and \(A_2\) are involutive matrices, to characterize all situations where a linear combination of the form (1) is a tripotent or an idempotent or an involutive matrix, then to determine all situations where a linear combination of the form (1) is an involutive matrix when \(A_1\) and \(A_2\) are tripotent or idempotent matrices.