Determine the characteristic polynomial of the matrix $$\begin{bmatrix}0&1\\1&0&1\\\space&1&\cdot&\cdot\\\space&\space&\cdot&\cdot&\cdot\\\space&\space\space&\space\space&\space&\cdot&1\\\space&\space&\space&\space&1&0\end{bmatrix}$$
Solution:
Let, $$A^n_{\lambda}=\begin{bmatrix}-\lambda&1\\1&-\lambda&1\\\space&1&\cdot&\cdot\\\space&\space&\cdot&\cdot&\cdot\\\space&\space\space&\space\space&\space&\cdot&1\\\space&\space&\space&\space&1&-\lambda\end{bmatrix}$$
I found the recursion relation: $$A^n_{\lambda}=-\lambda A^{n-1}_{\lambda}-A^{n-2}_{\lambda}$$
I don't know how to solve recursion relations. Can someone help me? Thanks.