Finding Eigenvectors and Eigenvalues of Symmetric Matrix Dimension n.

Question

Quite stuck with the following question:
Find the eigenvalues and eigenvectors of:
$\begin{bmatrix}-2 & 1 &0& ......&0\\1 & -2 & 1&......&0\\0&1&-2&......&.\\.&.&.&......&.\\.&.&.&......&1\\.&.&.&1&-2\end{bmatrix}$

Where the matrix is $n \times n$.

Problems:

I found the eigenvalues for the two and three dimensional case as being $\lambda= -1,-3$ and something different for the three dimensional case so I had no idea how to generalize to n dimensions. Any help would be appreciated.

Answer 1

Hint: let $d_n=\det (\lambda I_n -A_n)$. Computed by the first row is equal to $(\lambda+2)d_{n-1}-(-1)(-1)d_{n-2}=(\lambda+2)d_{n-1}-d_{n-2}$. You can finish?