Affiliation:
1. Department of Mathematics, Faculty of Science, Silpakorn University , Nakhon Pathom 73000 , Thailand
Abstract
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.
Reference14 articles.
1. R. P. Brent and B. D. McKay, Determinants and ranks of random matrices over Zm, Discrete Math. 66 (1987), 35–49.
2. L. Cao, D. McLaren, and S. Plosker, The complete positivity of symmetric tridiagonal and pentadiagonal matrices, Spec. Matrices 11 (2023), 20220173.
3. W. Cheney and D. R. Kincaid, Linear Algebra Theory and Applications, 2nd edition, Jones and Bartlett Learning, Sudbury, Massachusetts, 2012.
4. P. Choosuwan, S. Jitman, and P. Udomkavanich, Determinants of matrices over commutative finite principal idealrings, Finite Fields Appl. 48 (2017), 126–140.
5. C. M. da Fonseca, On the eigenvalues of some tridiagonal matrices, J. Comput. Appl. Math. 200 (2007), 283–286.