linear algebra - Find the determinant of the following;

Find the determinant of the following matrix, and for which value of $x$ is it invertible;
$$\begin{pmatrix}
x & 1 & 0 & 0 & 0 & \ldots & 0 & 0 \\
0 & x & 1 & 0 & 0 & \ldots & 0 & 0 \\
0 & 0 & x & 1 & 0 & \ldots & 0 & 0 \\
\ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\
\ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\
\ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\
0 & 0 & 0 & 0 & 0 & \ldots & x & 1 \\

1 & 0 & 0 & 0 & 0 & \ldots & 0 & x
\end{pmatrix}$$

Now I don't really know how to procees as I get find a suitable row operations that will simplify the process so I thought I would look at cases, maybe see a pattern.

$\mathbf{2 \times 2}$

$\begin{bmatrix}x & 1\\1 & x\end{bmatrix}$
This has determinant; $x^2-1$

$\mathbf{3 \times 3}$

$\begin{bmatrix}x & 1 & 0\\0 & x & 1\\1 & 0 & x\end{bmatrix}$
This has determinant $x^3+1$

So is that the pattern?
determinant is $x^n-1$ if $n$ is even,
determinant is $x^n+1$ if $n$ is odd??