If $\mathrm{A}=\left(\begin{array}{cc}\cos \mathrm{\theta }& \sin \mathrm{\theta }\\ -\sin \mathrm{\theta }& \cos \mathrm{\theta }\end{array}\right)$ where $\mathrm{\theta }=\frac{2\mathrm{\pi }}{19}$ then ${\mathrm{A}}^{2017}=$

$$\begin{gathered} {\mathrm{A}}^2=\left[\begin{array}{cc}\cos \mathrm{\theta }& \sin \mathrm{\theta }\\ -\sin \mathrm{\theta }& \cos \mathrm{\theta }\end{array}\right] \left[\begin{array}{cc}\cos \mathrm{\theta }& \sin \mathrm{\theta }\\ -\sin \mathrm{\theta }& \cos \mathrm{\theta }\end{array}\right] \\ =\left[\begin{array}{cc}\cos 2\mathrm{\theta }& \sin 2\mathrm{\theta }\\ -\sin 2\mathrm{\theta }& \cos 2\mathrm{\theta }\end{array}\right] \\ \mathrm{similarly}, {\mathrm{A}}^{2017}=\left[\begin{array}{cc}\cos 2017 \mathrm{\theta }& \sin 2017 \mathrm{\theta }\\ -\sin 2017 \mathrm{\theta }& \cos 2017 \mathrm{\theta }\end{array}\right] \end{gathered}$$

$$\begin{gathered} \Rightarrow {\mathrm{A}}^{2017}={\mathrm{A}}^3·{\mathrm{A}}^{2014} \\ \mathrm{since} \mathrm{\theta }=\frac{2\mathrm{\pi }}{19} \Rightarrow 2014 \mathrm{\theta }=2014\times \frac{2\mathrm{\pi }}{19} \\ =212\mathrm{\pi } \\ \therefore \mathrm{Los}(212 \mathrm{\pi })=1, \sin (212 \mathrm{\pi })=0 \\ \therefore {\mathrm{A}}^{2017}={\mathrm{A}}^3\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \\ ={\mathrm{A}}^3 \end{gathered}$$