if matrix cos2pi/7 -sin2pi/7

sin2pi/7 cos 2pi/7 the whole power k = matrix 1 0

0 1 , then write the value of x+y+xy

${\left[\begin{array}{cc}\cos \frac{2\mathrm{\pi }}{7}& -\sin \frac{2\mathrm{\pi }}{7}\\ \sin \frac{2\mathrm{\pi }}{7}& \cos \frac{2\mathrm{\pi }}{7}\end{array}\right]}^k=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Now, ${\left[\begin{array}{cc}\cos \frac{2\mathrm{\pi }}{7}& -\sin \frac{2\mathrm{\pi }}{7}\\ \sin \frac{2\mathrm{\pi }}{7}& \cos \frac{2\mathrm{\pi }}{7}\end{array}\right]}^2=\left[\begin{array}{cc}\cos \frac{2\mathrm{\pi }}{7}& -\sin \frac{2\mathrm{\pi }}{7}\\ \sin \frac{2\mathrm{\pi }}{7}& \cos \frac{2\mathrm{\pi }}{7}\end{array}\right]\times \left[\begin{array}{cc}\cos \frac{2\mathrm{\pi }}{7}& -\sin \frac{2\mathrm{\pi }}{7}\\ \sin \frac{2\mathrm{\pi }}{7}& \cos \frac{2\mathrm{\pi }}{7}\end{array}\right]$

$\Rightarrow {\left[\begin{array}{cc}\cos \frac{2\mathrm{\pi }}{7}& -\sin \frac{2\mathrm{\pi }}{7}\\ \sin \frac{2\mathrm{\pi }}{7}& \cos \frac{2\mathrm{\pi }}{7}\end{array}\right]}^2=\left[\begin{array}{cc}{\cos }^2\left(\frac{2\mathrm{\pi }}{7}\right)-{\sin }^2\left(\frac{2\mathrm{\pi }}{7}\right)& -2\sin \left(\frac{2\mathrm{\pi }}{7}\right)\cos \left(\frac{2\mathrm{\pi }}{7}\right)\\ 2\sin \left(\frac{2\mathrm{\pi }}{7}\right)\cos \left(\frac{2\mathrm{\pi }}{7}\right)& {\cos }^2\left(\frac{2\mathrm{\pi }}{7}\right)-{\sin }^2\left(\frac{2\mathrm{\pi }}{7}\right)\end{array}\right]$

$\Rightarrow {\left[\begin{array}{cc}\cos \frac{2\mathrm{\pi }}{7}& -\sin \frac{2\mathrm{\pi }}{7}\\ \sin \frac{2\mathrm{\pi }}{7}& \cos \frac{2\mathrm{\pi }}{7}\end{array}\right]}^2=\left[\begin{array}{cc}\cos \left(2\times \frac{2\mathrm{\pi }}{7}\right)& -\sin \left(2\times \frac{2\mathrm{\pi }}{7}\right)\\ \sin \left(2\times \frac{2\mathrm{\pi }}{7}\right)& \cos \left(2\times \frac{2\mathrm{\pi }}{7}\right)\end{array}\right]$

Again, $\Rightarrow {\left[\begin{array}{cc}\cos \frac{2\mathrm{\pi }}{7}& -\sin \frac{2\mathrm{\pi }}{7}\\ \sin \frac{2\mathrm{\pi }}{7}& \cos \frac{2\mathrm{\pi }}{7}\end{array}\right]}^3=\left[\begin{array}{cc}\cos \frac{2\mathrm{\pi }}{7}& -\sin \frac{2\mathrm{\pi }}{7}\\ \sin \frac{2\mathrm{\pi }}{7}& \cos \frac{2\mathrm{\pi }}{7}\end{array}\right]\times {\left[\begin{array}{cc}\cos \frac{2\mathrm{\pi }}{7}& -\sin \frac{2\mathrm{\pi }}{7}\\ \sin \frac{2\mathrm{\pi }}{7}& \cos \frac{2\mathrm{\pi }}{7}\end{array}\right]}^2$

$\Rightarrow {\left[\begin{array}{cc}\cos \frac{2\mathrm{\pi }}{7}& -\sin \frac{2\mathrm{\pi }}{7}\\ \sin \frac{2\mathrm{\pi }}{7}& \cos \frac{2\mathrm{\pi }}{7}\end{array}\right]}^3=\left[\begin{array}{cc}\cos \frac{2\mathrm{\pi }}{7}& -\sin \frac{2\mathrm{\pi }}{7}\\ \sin \frac{2\mathrm{\pi }}{7}& \cos \frac{2\mathrm{\pi }}{7}\end{array}\right]\times \left[\begin{array}{cc}\cos \left(2\times \frac{2\mathrm{\pi }}{7}\right)& -\sin \left(2\times \frac{2\mathrm{\pi }}{7}\right)\\ \sin \left(2\times \frac{2\mathrm{\pi }}{7}\right)& \cos \left(2\times \frac{2\mathrm{\pi }}{7}\right)\end{array}\right]$

$\Rightarrow {\left[\begin{array}{cc}\cos \frac{2\mathrm{\pi }}{7}& -\sin \frac{2\mathrm{\pi }}{7}\\ \sin \frac{2\mathrm{\pi }}{7}& \cos \frac{2\mathrm{\pi }}{7}\end{array}\right]}^3=\left[\begin{array}{cc}\cos \frac{2\mathrm{\pi }}{7}\cos \left(2\times \frac{2\mathrm{\pi }}{7}\right)-\sin \frac{2\mathrm{\pi }}{7}\sin \left(2\times \frac{2\mathrm{\pi }}{7}\right)& -\sin \frac{2\mathrm{\pi }}{7}\cos \left(2\times \frac{2\mathrm{\pi }}{7}\right)-\sin \left(2\times \frac{2\mathrm{\pi }}{7}\right)\cos \frac{2\mathrm{\pi }}{7}\\ \sin \frac{2\mathrm{\pi }}{7}\cos \left(2\times \frac{2\mathrm{\pi }}{7}\right)+\sin \left(2\times \frac{2\mathrm{\pi }}{7}\right)\cos \frac{2\mathrm{\pi }}{7}& \cos \frac{2\mathrm{\pi }}{7}\cos \left(2\times \frac{2\mathrm{\pi }}{7}\right)-\sin \frac{2\mathrm{\pi }}{7}\sin \left(2\times \frac{2\mathrm{\pi }}{7}\right)\end{array}\right]$

$\Rightarrow {\left[\begin{array}{cc}\cos \frac{2\mathrm{\pi }}{7}& -\sin \frac{2\mathrm{\pi }}{7}\\ \sin \frac{2\mathrm{\pi }}{7}& \cos \frac{2\mathrm{\pi }}{7}\end{array}\right]}^3=\left[\begin{array}{cc}\cos \left(3\times \frac{2\mathrm{\pi }}{7}\right)& -\sin \left(3\times \frac{2\mathrm{\pi }}{7}\right)\\ \sin \left(3\times \frac{2\mathrm{\pi }}{7}\right)& \cos \left(3\times \frac{2\mathrm{\pi }}{7}\right)\end{array}\right]$

Similarly, if we go on, we will find that 

${\left[\begin{array}{cc}\cos \frac{2\mathrm{\pi }}{7}& -\sin \frac{2\mathrm{\pi }}{7}\\ \sin \frac{2\mathrm{\pi }}{7}& \cos \frac{2\mathrm{\pi }}{7}\end{array}\right]}^k=\left[\begin{array}{cc}\cos \left(k\times \frac{2\mathrm{\pi }}{7}\right)& -\sin \left(k\times \frac{2\mathrm{\pi }}{7}\right)\\ \sin \left(k\times \frac{2\mathrm{\pi }}{7}\right)& \cos \left(k\times \frac{2\mathrm{\pi }}{7}\right)\end{array}\right]$

i.e., $\Rightarrow {\left[\begin{array}{cc}\cos \frac{2\mathrm{\pi }}{7}& -\sin \frac{2\mathrm{\pi }}{7}\\ \sin \frac{2\mathrm{\pi }}{7}& \cos \frac{2\mathrm{\pi }}{7}\end{array}\right]}^k=\left[\begin{array}{cc}\cos \frac{2k\mathrm{\pi }}{7}& -\sin \frac{2k\mathrm{\pi }}{7}\\ \sin \frac{2k\mathrm{\pi }}{7}& \cos \frac{2k\mathrm{\pi }}{7}\end{array}\right]$  -------------------(1)

But, it is given that 

${\left[\begin{array}{cc}\cos \frac{2\mathrm{\pi }}{7}& -\sin \frac{2\mathrm{\pi }}{7}\\ \sin \frac{2\mathrm{\pi }}{7}& \cos \frac{2\mathrm{\pi }}{7}\end{array}\right]}^k=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ ------------------(2)

From (1) and (2), we have:-

$\left[\begin{array}{cc}\cos \frac{2k\mathrm{\pi }}{7}& -\sin \frac{2k\mathrm{\pi }}{7}\\ \sin \frac{2k\mathrm{\pi }}{7}& \cos \frac{2k\mathrm{\pi }}{7}\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

We are required to find the least positive integral value of k for which this holds true.

So, $\cos \frac{2k\mathrm{\pi }}{7}=1, and, \sin \frac{2k\mathrm{\pi }}{7}=0$

$$\begin{gathered} \Rightarrow 2k\mathrm{\pi }=0 \\ \Rightarrow \mathrm{k}=0 \end{gathered}$$

So, 0 is the least positive integral value of k.