The determinant, spectral properties, and inverse of a tridiagonal $k$-Toeplitz matrix over a commutative ring

A square matrix is $k$-Toeplitz if its diagonals are periodic sequences of period $k$. We find universal formulas for the determinant, the characteristic polynomial, some eigenvectors, and the entries of the inverse of any tridiagonal $k$-Toeplitz matrix (in particular, of any tridiagonal matrix) over any commutative unital ring, expressed in terms of the elementary operations of the ring. The results are proven using combinatorial identities and elementary linear algebra. We conduct a complexity analysis of algorithms based on our formulas, showing that they are efficient, and we compare our results favourably with those found in the literature. Concretely, the determinant, the characteristic polynomial, and any entry of the inverse of a tridiagonal $k$-Toeplitz matrix of size $n$ can each be found with $O(\displaystyle\log\frac nk+k)$ operations, while an eigenvector can be determined with $O(n+k)$ operations.