Question

If \[ A = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \] and \(\theta = \frac{2\pi}{7}\), then \(A^{100} = A \times A \times \ldots\) (100 times) is equal to:

• A. \(\begin{pmatrix} \cos 2\theta & -\sin 2\theta \\ \sin 2\theta & \cos 2\theta \end{pmatrix}\)
• B. \(\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\)
• C. \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)
• D. \(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\)

Hint:

When multiplying rotation matrices, the resulting angle is the sum of the individual angles of rotation. For large powers, use periodicity to simplify the angle.

Answer 1

The Correct Option is A

The matrix \( A \) is a 2D rotation matrix. Repeated multiplication of \( A \) rotates by \( \theta \) each time:

\[ A^k = \begin{pmatrix} \cos(k\theta) & -\sin(k\theta) \\ \sin(k\theta) & \cos(k\theta) \end{pmatrix}. \]

Therefore, \( A^{100} \) represents a rotation by \( 100\theta \):

\[ A^{100} = \begin{pmatrix} \cos(100\theta) & -\sin(100\theta) \\ \sin(100\theta) & \cos(100\theta) \end{pmatrix}. \]

Step 1: Simplify \( 100\theta \mod 2\pi \).

Given \( \theta = \frac{2\pi}{7} \), we have:

\[ 100\theta = 100 \cdot \frac{2\pi}{7} = \frac{200\pi}{7}. \]

Reduce \( \frac{200\pi}{7} \) modulo \( 2\pi \). Divide 200 by 7:

\[ 200 \div 7 = 28 \quad (\text{remainder } 4). \]

Hence:

\[ 100\theta = \frac{200\pi}{7} = 28 \cdot 2\pi + \frac{8\pi}{7}. \]

Modulo \( 2\pi \), this simplifies to:

\[ 100\theta \equiv \frac{8\pi}{7} \pmod{2\pi}. \]

Step 2: Express \( \frac{8\pi}{7} \) in terms of \( \theta \).

Since \( \theta = \frac{2\pi}{7} \):

\[ 100\theta \equiv 4\theta \pmod{2\pi}. \]

Step 3: Compute \( A^{100} \).

Using the formula for \( A^k \):

\[ A^{100} = \begin{pmatrix} \cos(4\theta) & -\sin(4\theta) \\ \sin(4\theta) & \cos(4\theta) \end{pmatrix}. \]

Substitute \( 4\theta \) into the matrix:

\[ A^{100} = \begin{pmatrix} \cos(2\theta) & -\sin(2\theta) \\ \sin(2\theta) & \cos(2\theta) \end{pmatrix}. \]

Conclusion: The result for \( A^{100} \) is:

\[ \begin{pmatrix} \cos(2\theta) & -\sin(2\theta) \\ \sin(2\theta) & \cos(2\theta) \end{pmatrix}. \]