If [(cos(2π/7),-sin(2π/7))(sin(2π/7),cos(2π/7))]^k =[(1,0)(0,1)], then the least positive integral value of k is : A. 3 B. 4 C. 6 D. 7 ​

Question

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

A. 3 

B. 4 

C. 6 

D. 7

Answer 1

(D). 7

\(\begin{bmatrix} cos\frac{2\pi}{7} & -sin\frac{2\pi}{7} \\[0.3em] sin\frac{2\pi}{7} & cos\frac{2\pi}{7} \\[0.3em] \end{bmatrix}^k\)= \(\begin{bmatrix} 1 & 0 \\[0.3em] 0 & 1 \\[0.3em] \end{bmatrix}\) 

So, 

k = 7 

A⁷ = I

Hence, 

k = 7. 

Option (D) is the correct answer.