Trace of the Positive Integer Powers (n-1)-Tridiagonal Toeplitz Matrix n×n

The trace of a matrix is obtained by summing the elements along the main diagonal of a square matrix. The matrix used in this study is a Toeplitz (n-1)-tridiagonal matrix of order n×n. The aim of this research is to determine the general form or formula for the trace of a Toeplitz (n-1)-tridiagonal matrix of order n×n raised to a positive integer power. This research is quantitative, with the research instrument being the collection of data from the multiplication of Toeplitz (n-1)-tridiagonal matrices starting from order 3×3 from powers 2 through 10. This process continues up to order 6×6 from powers 2 through 10, until the pattern becomes apparent. The results of the research are two general forms of the powers of the Toeplitz (n-1)-tridiagonal matrix of order n×n: one for odd positive integer powers and another for even positive integer powers, both of which have been proven using mathematical induction. Furthermore, by using the definition of the trace of a matrix obtained two general forms for the trace of the Toeplitz (n-1)-tridiagonal matrix of order n×n are also derived: one for odd positive integer powers and another for even positive integer powers from the general form of the matrix power. Given the application of these two general forms in example problems with the order 8x8 for powers 12 and 21.