Algebraic Aspects of combined matrices

In this work, we present algebraic results concerning the combined matrices $\mathcal{C}(A)$, where the entries of $A$ belong to a number field $K$ and $A$ is a non-singular matrix. In other words, $A$ is a $n\times n$ matrix belonging to the General Linear Group over $K$, denoted by $\mathrm{GL}_n(K)$. We also analyze the case in which matrix $A$ belongs to algebraic subgroups of $\mathrm{GL}_n(K)$, such as the unimodular group, where $A^2$ is a $n\times n$ matrix belonging to the Special Linear Group, denoted by $\mathrm{SL}_n(K)$, triangular groups, diagonal groups, among others. In particular, we thouroughly examine the cases $n=2$ and $n=3$ for symmetric and non-symmetric matrices, providing explicit diagonalization of $\mathcal{C}(A)$, which includes characteristic polynomials with their eigenvalues and eigenfactors.