# 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

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

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

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

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

Similarly, if we go on, we will find that

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

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

But, it is given that

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

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

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

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

So,

$⇒2k\mathrm{\pi }=0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒\mathrm{k}=0$

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

• -15

sry sry the ques is wrong... instead of "x+y+xy", "least positive inteagral value of k" has to come..

• 3
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