On Some Properties of Matrices with Entries Defined by Products of $k$-Fibonacci and $k$-Lucas Numbers

Abstract: In this paper, we employ combinatorial and algebraic tools to derive closed-form expressions for several classical matrix invariants, including the determinant, inverse, trace, and powers, for a family of matrices whose entries are given by products of $k$-Fibonacci and $k$-Lucas numbers. Moreover, we compute the spectral radius and the energy of the graphs associated with this family of matrices. Finally, we investigate connections between the obtained formulas and certain integer sequences listed in the On-Line Encyclopedia of Integer Sequences (OEIS).