Determinant, inertia, and inverse of a class of irreducible tridiagonal matrices

Abstract In this paper, we study a class of irreducible tridiagonal matrices and analyze their fundamental properties. We derive explicit formulas for their determinant, providing insights into their spectral behavior. Additionally, we establish results concerning their inertia and obtain closed-form expressions for their inverse. These findings contribute to the broader understanding of structured matrices and have potential applications in various fields of mathematics and other sciences.