If cos⁡2π7−sin⁡2π7sin⁡2π7cos⁡2π7k=1 00 1, then the least positive integral value of k, is

Solution:

We have,

$\begin{array}{l} {[\begin{array}{ll} \cos \frac{2 π}{7} & - \sin \frac{2 π}{7} \\ \sin \frac{2 π}{7} & \cos \frac{2 π}{7} \end{array}]}^{2} \\ = [\begin{array}{ll} \cos \frac{2 π}{7} & - \sin \frac{2 π}{7} \\ \sin \frac{2 π}{7} & \cos \frac{2 π}{7} \end{array}] [\begin{array}{ll} \cos \frac{2 π}{7} & - \sin \frac{2 π}{7} \\ \sin \frac{2 π}{7} & \cos \frac{2 π}{7} \end{array}] \\ = [\begin{matrix} \cos^{2} \frac{2 π}{7} - \sin^{2} \frac{2 π}{7} & - 2 \sin \frac{2 π}{7} \cos \frac{2 π}{7} \\ 2 \sin \frac{2 π}{7} \cos \frac{2 π}{7} & - \sin^{2} \frac{2 π}{7} + \cos^{2} \frac{2 π}{7} \end{matrix}] \end{array}$

$= [\begin{array}{cc} \cos \frac{4 π}{7} & - \sin \frac{4 π}{7} \\ \sin \frac{4 π}{7} & \cos \frac{4 π}{7} \end{array}]$

Continuing in this manner, we get

${[\begin{array}{rr} \cos \frac{2 π}{7} - \sin \frac{2 π}{7} \\ \sin \frac{2 π}{7} \cos \frac{2 π}{7} \end{array}]}^{7} = [\begin{array}{rr} \cos 2 π - \sin 2 π \\ \sin 2 π \cos 2 π \end{array}] = [\begin{array}{ll} 1 0 \\ 0 1 \end{array}]$

Hence, the least value of k is 7.

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

Solution:

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