Abstract: Firstly, by using the basic properties of generalized quadratic matrices, we studied the existence of the nontrivial solution (ρ,σ) for the linear combination ρA+σP of  generalized quadratic matrix A and an idempotent matrix P expressed as A²=αA+βP. The results show that when η²=4β+α²≠0, ρA+σP has only two nontrivial solutions, and the matrix A can be uniquely expressed as a linear combination of idempotent matrix generated by these two nontrivial solutions. Secondly, we discussed the case for nontrivial solution of ρA+σP when η²=4β+α²=0.

Key words: generalized quadratic matrix, idempotent matrix, nilpotent matrix, linear combination, nontrivial solution