Determinants of tridiagonal matrices over some commutative finite chain rings

Abstract

Diagonal matrices and their generalization in terms of tridiagonal matrices have been of interest due to their nice algebraic properties and wide applications. In this article, the determinants of tridiagonal matrices over a finite field Fq{{\mathbb{F}}}_{q} and a commutative finite chain ring RR are studied. The main focus is the enumeration of tridiagonal matrices with prescribed determinant. The number of tridiagonal matrices with prescribed determinant over Fq{{\mathbb{F}}}_{q} and the number of non-singular tridiagonal matrices with prescribed determinant over RR are completely determined. For singular tridiagonal matrices with prescribed determinant over RR, bounds on the number of such matrices with prescribed determinant are given. Subsequently, the number of some special tridiagonal matrices with prescribed determinant over Fq{{\mathbb{F}}}_{q} and RR is presented.

Keywords

• tridiagonal matrices
• determinants
• finite fields
• finite commutative chain rings
• enumeration
• 11c20
• 15b33