Determinants of Tridiagonal and Circulant Matrices Special Form by Chebyshev Polynomials

Nurliantika Nurliantika; Fransiskus Fran; Yundari Yundari

doi:10.31764/jtam.v9i1.27871

Determinants of Tridiagonal and Circulant Matrices Special Form by Chebyshev Polynomials

Nurliantika Nurliantika, Fransiskus Fran, Yundari Yundari

Abstract

Along with the development of science, many researchers have found new methods to determine the determinant of a matrix of more than three orders. Chebyshev polynomial can be used to find and develop a more efficient formula in calculating the determinant of matrices. This research explores the Chebyshev polynomials of the first kind T_n (x) and second kind U_n (x). Both types of Chebyshev polynomials, T_n (x) and U_n (x), can be represented using recurrence relations. This research aims to determine the determinant of tridiagonal and circulant matrices of special form using Chebyshev polynomials T_n (x) and U_n (x). Determining the determinant of a matrix is a fundamental problem in linear algebra that plays an important role in both theoretical and applied mathematics. Its theoretical contributions include a deeper understanding of matrix properties, the development of efficient computational methods, and the explanation of the relationship between matrices and orthogonal polynomials. By utilizing Chebyshev polynomials, this study strengthens determinant theory, particularly for matrices with special shapes. The steps to determine the determinant of tridiagonal and circulant matrices involve the application of elementary row operations. The first step is to perform row operations on the tridiagonal and circulant matrices to obtain a matrix form that conforms to the determinant theorem of the tridiagonal and circulant matrices. After the elementary row operation is applied, if the form of the tridiagonal and circulant matrices each satisfies the form in the determinant theorem of the tridiagonal and circulant matrices, then the determinant of the matrices can be calculated using each of the theorems that satisfy. Then the determinants of the tridiagonal and the circulant matrices are obtained. The results of this study show that the determinant of tridiagonal and circulant matrices of special form can be determined using Chebyshev polynomials T_n (x) and U_n (x).

Keywords

Chebyshev Polynomial of the First Kind (T_n (x)); Chebyshev Polynomial of the Second Kind (U_n (x)); Recurrence Relation.

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DOI: https://doi.org/10.31764/jtam.v9i1.27871