Question

If $A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$, then the matrix $A^{-50}$ when $\theta =\frac{\pi }{12}$ , is equal to :

• A. $\left[\begin{array}{cc}\frac{\sqrt{3}}{2}& \frac{1}{2}\\ -\frac{1}{2}& \frac{\sqrt{3}}{2}\end{array}\right]$
• B. $\left[\begin{array}{cc}\frac{1}{2}& \frac{\sqrt{3}}{2}\\ -\frac{\sqrt{3}}{2}& \frac{1}{2}\end{array}\right]$
• C. $\left[\begin{array}{cc}\frac{1}{2}& -\frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}& \frac{1}{2}\end{array}\right]$
• D. $\left[\begin{array}{cc}\frac{\sqrt{3}}{2}& -\frac{1}{2}\\ \frac{1}{2}& \frac{\sqrt{3}}{2}\end{array}\right]$

Answer 1

Answer: (A)

Here, $A{A}^T=I$
$\Rightarrow A^{-1}=A^T=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$
$A^{-n}=\left[\begin{array}{cc}\cos \left(n\theta \right)& \sin \left(n\theta \right)\\ -\sin \left(n\theta \right)& \cos \left(n\theta \right)\end{array}\right]$
$\therefore A^{-50}=\left[\begin{array}{cc}\cos \left(50\right)\theta & \sin \left(50\right)\theta \\ -\sin \left(50\right)\theta & \cos \left(50\right)\theta \end{array}\right]$
$=\left[\begin{array}{cc}\frac{\sqrt{3}}{2}& \frac{1}{2}\\ -\frac{1}{2}& \frac{\sqrt{3}}{2}\end{array}\right]$