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 F q {{\mathbb{F}}}_{q} and a commutative finite chain ring R R are studied. The main focus is the enumeration of tridiagonal matrices with prescribed determinant. The number of tridiagonal matrices with prescribed determinant over F q {{\mathbb{F}}}_{q} and the number of non-singular tridiagonal matrices with prescribed determinant over R R are completely determined. For singular tridiagonal matrices with prescribed determinant over R R , bounds on the number of such matrices with prescribed determinant are given. Subsequently, the number of some special tridiagonal matrices with prescribed determinant over F q {{\mathbb{F}}}_{q} and R R is presented.