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

TL;DR

This paper investigates matrices with entries formed by products of $k$-Fibonacci and $k$-Lucas numbers, deriving explicit formulas for invariants, spectral properties, and connections to integer sequences.

Contribution

It provides unified formulas for matrix invariants and spectral properties of matrices based on $k$-Fibonacci and $k$-Lucas numbers, revealing new algebraic patterns.

Findings

Explicit formulas for determinant, inverse, trace, and powers of the matrices.

Determination of spectral radius and graph energy related to these matrices.

Connections established between matrix properties and OEIS integer sequences.

Abstract

In this paper, we study a structured family of matrices whose entries are given by products of $k$-Fibonacci and $k$-Lucas numbers. For this family, we obtain explicit and unified formulas for several classical matrix invariants, including the determinant, inverse, trace, and matrix powers, revealing nontrivial algebraic patterns induced by the underlying recurrence relations. In addition, we determine the spectral radius and the energy of the graphs naturally associated with these matrices. Finally, we establish connections between the resulting formulas and certain integer sequences recorded in the On-Line Encyclopedia of Integer Sequences (OEIS).