Question

If \(A = [\begin{array}{cc}cosθ & -sinθ \\ sinθ & cosθ\end{array}]\), then the matrix \(A^{-3}\) when \(θ = \frac{π}{6}\) is equal to...

• A. \([\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}]\)
• B. \([\begin{array}{cc}0 & 1 \\ 1 & 0\end{array}]\)
• C. \([\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}]\)
• D. \([\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}]\)

Hint:

A is a rotation matrix, so A^n rotates by n theta.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The matrix \(A=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}\) is a rotation matrix. Multiplying such matrices adds the angles, so \(A^n\) is the rotation by \(n\theta\), and \(A^{-1}\) is the rotation by \(-\theta\).

Step 2: Apply:
\(A^{-3}\) is the rotation by \(-3\theta=-3\times\dfrac\pi6=-\dfrac\pi2\).

Step 3: Write the matrix:
\[ A^{-3}=\begin{pmatrix}\cos(-\frac\pi2)&-\sin(-\frac\pi2)\\\sin(-\frac\pi2)&\cos(-\frac\pi2)\end{pmatrix}=\begin{pmatrix}0&1\\-1&0\end{pmatrix} \]

Step 4: Choose:
Option (A). Option (C) is the rotation by \(+\tfrac\pi2\), which would be \(A^{3}\).

Final Answer:
A^-3 is a rotation by -90 degrees. \[ \boxed{\begin{pmatrix}0&1\\-1&0\end{pmatrix}} \]