New Proofs of Binet’s Formulas Using The Lucas Q-Matrix

Abstract

The Fibonacci and Lucas sequences play a fundamental role in Number Theory and have deep connections with Linear Algebra and Matrix Theory. This paper explores an alternative approach to proving Binet’s formulas for these sequences using the Lucas -matrix, denoted by . Introduced in 2010 by Köken and Bozkurt, this matrix exhibits an intriguing alternating behavior: for even exponents, its powers relate solely to Fibonacci numbers, while for odd exponents, they involve Lucas numbers. Despite its potential, the -matrix remains relatively unexplored in the literature. We present two novel proofs of Binet’s formulas based on matrix algebra. The first proof leverages the spectral properties of , employing its characteristic polynomial, eigenvalues, and eigenvectors to derive explicit expressions for its powers. The second proof follows a direct computation of the characteristic polynomial of , establishing its connection with Fibonacci and Lucas sequences. These methods not only reinforce classical results but also reveal the structural elegance of , highlighting its role as a bridge between recurrence relations and linear transformations.