Let X_i, i=1,2,...,m, be diagonalizable matrices that mutually commute. This paper provides a combinatorial method to handle the problem of when a linear combination matrix X=\sum_{i=1}^{m}c_iX_i is a matrix such that \sigma(X)\subseteq\{\lambda_1, \lambda_2,..., \lambda_{n}\}, where c_i, i=1,2,...,m, are nonzero complex scalars and \sigma(X) denotes the spectrum of the matrix X. If the spectra of the matrices X and X_i, i=1,2,...,m, are chosen as subsets of some particular sets, then this problem is equivalent to the problem of characterizing all situations in which a linear combination of some commuting special types of matrices, e.g. the matrices such that A^k=A, k=2,3,..., is also a special type of matrix. The method developed in this note makes it possible to solve such characterization problems for the linear combinations of finitely many special types of matrices. Moreover, the method is illustrated by considering the problem, which is one of the open problems left in [Linear Algebra Appl. 437 (2012) 2091-2109], of characterizing all situations in which a linear combination X=c_1X_1+c_2X_2+c_3X_3 is a tripotent matrix when X_1 is an involutory matrix and both X_2 and X_3 are tripotent matrices that mutually commute. The results obtained cover those established in the reference above.