The generalized quadraticity of linear combinations of two commuting quadratic matrices

Let A(1) and A(2) be two nonzero commuling} and {a2, fi2}-quadratic matrices, respectively, where alpha(1), beta(1), alpha(2), beta(2) is an element of C with alpha(1) not equal beta(1) and alpha(2) not equal beta(2). The aim of this work is mainly to characterize all situations, where the linear combination a(1)A(1) + a(2)A(2) is a generalized quadratic matrix. The results established here cover many of the results in the literature related to idempotency, in volutivi ty and tripotency of the linear combinations of idempotent and/or in volutive matrices.