Explicit Determinantal Formula for a Class of Banded Matrices

Abstract

This short note presents a concise proof of a recently proposed explicit formula for the determinant of a particular family of banded (pentadiagonal) matrices, structured with parameters a and b and within a Hessenberg-like layout. The formula expresses det Aₙ in terms of congruence classes of n modulo 4, with closed forms: (a−1)² if n≡0, a² + b + 1 if n≡1, a² + 2a−b if n≡2, and a² if n≡3.