If ${\left[\begin{array}{cc}\cos \frac{2\mathrm{\pi }}{7}& -\sin \frac{2\mathrm{\pi }}{7}\\ \sin \frac{2\mathrm{\pi }}{7}& \cos \frac{2\mathrm{\pi }}{7}\end{array}\right]}^{\mathrm{k}}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then the least positive integral value of k is

we know that ${\left[\begin{array}{cc}\cos \mathrm{\theta }& -\sin \mathrm{\theta }\\ \sin \mathrm{\theta }& \cos \mathrm{\theta }\end{array}\right]}^{\mathrm{n}}=\left[\begin{array}{cc}\cos \mathrm{n\theta }& -\sin \mathrm{n\theta }\\ \sin \mathrm{n\theta }& \cos \mathrm{n\theta }\end{array}\right]$

$\begin{array}{c}\therefore {\left[\begin{array}{cc}\cos \frac{2\mathrm{\pi }}{7}& -\sin \frac{2\mathrm{\pi }}{7}\\ \sin \frac{2\mathrm{\pi }}{7}& \cos \frac{2\mathrm{\pi }}{7}\end{array}\right]}^{\mathrm{k}}=\left[\begin{array}{cc}\cos \frac{2\mathrm{k\pi }}{7}& -\sin \frac{2\mathrm{k\pi }}{7}\\ \sin \frac{2\mathrm{k\pi }}{7}& \cos \frac{2\mathrm{k\pi }}{7}\end{array}\right]\\ =\left[\begin{array}{ll}1& 0\\ 0& 1\end{array}\right],\text{ if }\mathrm{k}=7\mathrm{m},\text{ where }\mathrm{m}\in \mathrm{N}\end{array}$

Hence, the least value of k is 7.