Characterizing of quadraticity of linear combinations of two cubic matrices that commute

In the first chapter, some preliminary information about the study is given and it is emphasized the importance of idempotent matrices. Moreover, it is mentioned from some studies existing in the literature related to matrix classes discussed in the study. In the third chapter of the study, the problem of changing the quadraticity of the linear combination of two cubic matrices is discussed. While doing this, first a cubic matrix is expressed in terms of a generalized quadratic matrix, and then this generalized quadratic matrix is divided into submatrices in terms of quadratic matrix, and these submatrices are expressed in terms of idempotent matrices. Then, the necessary and sufficient conditions for the linear composition matrix of these matrices to be a quadratic matrix are obtained by using block matrices and elementary matrix operations. It can be seen that the results obtained cover many results in the literature regarding the characterization of linear combinations of special types of matrices. The last chapter consists of some discussions and suggestions related to the study.