Does anyone have an idea how to prove the following identity? 

$$
\mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix}  
x^{-2j} & -x^{2j+1} \\
1 & 0
\end{pmatrix}\right)=
\begin{cases}
2 & \text{if } n=0\pmod{6}\\
1 & \text{if } n=1,5\pmod{6}\\
-1 & \text{if } n=2,4\pmod{6}\\
4 & \text{if } n=3\pmod{6}
\end{cases},
$$
where $x=e^{\frac{\pi i}{n}}$ and the product sign means usual matrix multiplication.

I have tried induction but there are too many terms in all of four entries as $n$ grows. I think maybe using generating functions is the way?