Question

The characteristic roots of the two rowed orthogonal matrix $\left[\begin{array}{cc}\cos \theta & \sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$ are $\lambda$ and $\overline{\lambda }$. ($\lambda$ lies in $I$ quadrant for $\theta \in (0,{90}^o))$, then ${\lambda }^5$ is equal to

• A. $\cos 5\theta -i\sin 5\theta$
• B. $\cos 5\theta +i\sin 5\theta$
• C. ${\cos }^5\theta -i{\sin }^5\theta$
• D. ${\cos }^5\theta +i{\sin }^5\theta$

Answer 1

The correct option is B $\cos 5\theta +i\sin 5\theta$

We have,
$|A-\lambda I|=\left[\begin{array}{cc}\cos \theta -\lambda & -\sin \theta \\ \sin \theta & \cos \theta -\lambda \end{array}\right]$

$=(\cos \theta -\lambda {)}^2+{\sin }^2\theta$

Therefore, characteristic equation of $A$ is
$(\cos \theta -\lambda {)}^2+{\sin }^2\theta =0$

or $\cos \theta -\lambda =\pm i\sin \theta$

or $\lambda =\cos \theta \pm i\sin \theta$ which are of unit modulus.
But given $\lambda$ lies in first quadrant,

$\Rightarrow \lambda =\cos \theta +i\sin \theta$

$\therefore {\lambda }^5=\cos 5\theta +i\sin 5\theta$