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The generalized quadraticity of linear combinations of two commuting quadratic matrices

Let and be two nonzero commuting and -quadratic matrices, respectively, where with and . The aim of this work is mainly to characterize all situations, where the linear combination is a generalized quadratic matrix. The results established here cover many of the results in the literature related to idempotency, involutivity and tripotency of the linear combinations of idempotent and/or involutive matrices.