📘 Question
If
\[
\left[
\begin{array}{cc}
\cos\frac{2\pi}{7} & -\sin\frac{2\pi}{7} \\
\sin\frac{2\pi}{7} & \cos\frac{2\pi}{7}
\end{array}
\right]^k
=
\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
\]
Find the least positive integer value of \(k\).
✏️ Step-by-Step Solution
Step 1: Recognize the matrix
This is a rotation matrix with angle:
\[
\theta = \frac{2\pi}{7}
\]
Step 2: Property of rotation matrices
\[ A^k = \begin{bmatrix} \cos(k\theta) & -\sin(k\theta) \\ \sin(k\theta) & \cos(k\theta) \end{bmatrix} \]
Step 3: Condition for identity matrix
For identity matrix:
\[
\cos(k\theta) = 1 \quad \text{and} \quad \sin(k\theta) = 0
\]
This happens when:
\[
k\theta = 2\pi
\]
Step 4: Solve for \(k\)
\[
k \cdot \frac{2\pi}{7} = 2\pi
\]
\[
k = 7
\]
✅ Final Answer
\[
\boxed{7}
\]
💡 Key Concept
A rotation matrix returns to identity after a full rotation \(2\pi\). So, \(k\theta = 2\pi\) gives the smallest positive \(k\).