linear algebra - find a recursive relation for the characteristic polynomial of the $k times k $ matrix?

find a recursive relation for the characteristic polynomial of the $k \times k $ matrix $$\begin{pmatrix} 0 & 1 \\ 1 & 0 & 1 \\ \mbox{ } & 1 & . & . \\ \mbox{ } &\mbox{ } & . & . &. & \mbox{ } \\ \mbox{ } &\mbox{ } &\mbox{ } & . & . & 1 \\ \mbox{ } & \mbox{ } & \mbox{ } & \mbox{ } & 1 & 0\end{pmatrix}$$

and compute the polynomial for $k\le 5$

My attempt : Let $M_k$ be the $k\times k$ matrix and $P_k(x)=\det(M_k-xI_k)$ be its characteristic polynomial. We have

$$P_k(x)=\det\begin{pmatrix} -x & 1 \\ 1 & -x & 1 \\ \mbox{ } & 1 & \ddots & \ddots \\ \mbox{ } &\mbox{ } & \ddots & . &. & \mbox{ } \\ \mbox{ } &\mbox{ } &\mbox{ } & . & . & 1 \\ \mbox{ } & \mbox{ } & \mbox{ } & \mbox{ } & 1 & -x\end{pmatrix}$$

after that im not able proceed further

Any hints/solution will be appreciated

Answer

Hint: Look at the formula for the determinant of using the first row.
Then you get a recursive definition for the determinant.

Solution:

$P_k(x)=-xP_{k-1}(x)-P_{k-1}(-1)$