  In recent years, a number of fast algorithms for computing the determinant of
a Toeplitz matrix were developed. The fastest algorithm we know so far is of
order $k^2\log{n}+k^3$, where $n$ is the number of rows of the Toeplitz matrix
and $k$ is the bandwidth size. This is possible because such a determinant can
be expressed as the determinant of certain parts of $n$-th power of a related
$k \times k$ companion matrix. In this paper, we give a new elementary proof of
this fact, and provide various examples. We give symbolic formulas for the
determinants of Toeplitz matrices in terms of the eigenvalues of the
corresponding companion matrices when $k$ is small.
