Abstract: In this article, the determinant of tridiagonal Toeplitz matrices is determined recursively and explicitly. The method used is descriptive exploratory the journal written by Fitri Aryani. The inverse of tridiagonal Toeplitz matrices is calculated using the adjoint method, but the determinant and adjoint of the matrices are based on the recursive calculation of the determinant. With this approach, the formulas for the determinant and inverse of tridiagonal Toeplitz matrices can be formulated clearly and efficiently. This study demonstrates the effectiveness of the method used in simplifying computations and provides an algorithm for the formulation.

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.

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.

Abstract: A novel factorization for the sum of two single-pair matrices is established as product of lower-triangular, tridiagonal, and upper-triangular matrices, leading to semi-closed-form formulas for tridiagonal matrix inversion. Subsequent factorizations are established, leading to semi-closed-form formulas for the inverse sum of two single-pair matrices. An application to derive the symbolic inverse of a particular Gram matrix is presented, and the numerical stability of the formulas is studied.