The inverse of a tridiagonal $k$-Toeplitz matrix

An square matrix is $k$-Toeplitz if its diagonals are periodic sequences of period $k$. We find rational formulas for the determinant, the characteristic polynomial, and the elements of the inverse of a tridiagonal $k$-Toeplitz matrix (in particular, of any tridiagonal matrix) over any commutative unital ring, using only elementary linear algebra.