Determinant of a special matrix The question is related to the eigenvalue problem. Using MAPLE, I calculated the following determinants:
$$\begin{align}
\begin{vmatrix}
   -\lambda & 1 \\
  1 & -\lambda \\
\end{vmatrix}&=\lambda^2-1\\
\begin{vmatrix}
  -\lambda & 1 & 0 \\
   1& -\lambda & 1 \\
   0& 1 & -\lambda \\
\end{vmatrix}&=-\lambda^3+2\lambda\\
\begin{vmatrix}
  -\lambda & 1 &0  &1  \\
   1& -\lambda & 1 &0  \\
   0&  1& -\lambda & 1 \\
   1&  0&  1& -\lambda \\
\end{vmatrix}&=\lambda^4-4\lambda^2\\
\begin{vmatrix}
  -\lambda &1  &0  &1  &0  \\
   1& -\lambda &1  &0  &1  \\
   0&  1& -\lambda &1  &0  \\
   1&  0&  1& -\lambda &1  \\
   0&  1&  0&  1& -\lambda \\
\end{vmatrix}&=-\lambda^5+6\lambda^3\\
\begin{vmatrix}
  -\lambda &1  &0  &1  &0  &1  \\
   1& -\lambda &1  &0  &1  &0  \\
   0&  1& -\lambda &1  &0  &1  \\
   1&  0&  1& -\lambda &1  &0  \\
   0&  1&  0&  1& -\lambda &1  \\
   1&  0&  1&  0&1 & -\lambda \\
\end{vmatrix}&=-9\lambda^4+\lambda^6\\
\end{align}
$$
But I have no idea how to calculate the determinants quickly by hand. Here is my question:  

What is the determinant in the $n$ by $n$ case?


Based on the given results, you should be able to guess what the eigenvalues are. Can you write down an explicit set of eigenvectors for those eigenvalues? Does your construction generalize? 
