If \( \begin{bmatrix} cos\,\frac{2\pi}{7} & -sin\,\frac{2\pi}{7} \\[0

1.	If \( \begin{bmatrix} cos\,\frac{2\pi}{7} & -sin\,\frac{2\pi}{7} \\[0.3em] sin\,\frac{2\pi}{7} & cos\,\frac{2\pi}{7} \end{bmatrix}^k\) = \( \begin{bmatrix} 1 &0 \\[0.3em] 0 & 1 \end{bmatrix}\),then find the least positive integral value of k.
Answer» Least positive integral value of k is 7. Since we have, \( \begin{bmatrix} cos\,\frac{2\pi}{7} & -sin\,\frac{2\pi}{7} \\[0.3em] sin\,\frac{2\pi}{7} & cos\,\frac{2\pi}{7} \end{bmatrix}^7\) =\( \begin{bmatrix} cos\,2\pi & -sin\,2\pi \\[0.3em] sin\,2\pi & cos\,2\pi \end{bmatrix}\) = \( \begin{bmatrix} 1 & 0\\[0.3em] 0 & 1 \end{bmatrix}\)

If \( \begin{bmatrix} cos\,\frac{2\pi}{7} & -sin\,\frac{2\pi}{7} \\[0.3em] sin\,\frac{2\pi}{7} & cos\,\frac{2\pi}{7} \end{bmatrix}^k\) = \( \begin{bmatrix} 1 &0 \\[0.3em] 0 & 1 \end{bmatrix}\),then find the least positive integral value of k.

Answer»

Least positive integral value of k is 7.

Since we have,

\( \begin{bmatrix} cos\,\frac{2\pi}{7} & -sin\,\frac{2\pi}{7} \\[0.3em] sin\,\frac{2\pi}{7} & cos\,\frac{2\pi}{7} \end{bmatrix}^7\)

=\( \begin{bmatrix} cos\,2\pi & -sin\,2\pi \\[0.3em] sin\,2\pi & cos\,2\pi \end{bmatrix}\)

= \( \begin{bmatrix} 1 & 0\\[0.3em] 0 & 1 \end{bmatrix}\)