Question

If AB = 0, then for the matrices $A=\left[\begin{array}{cc}{\cos }^2\theta & \cos \theta \sin \theta \\ \cos \theta \sin \theta & {\sin }^2\theta \end{array}\right]andB=\left[\begin{array}{cc}{\cos }^2\varphi & \cos \varphi \sin \varphi \\ \cos \varphi \sin \varphi & {\sin }^2\varphi \end{array}\right],\theta -\varphi$ is

• A. an odd muliple of $\frac{\pi }{2}$
• B. an odd multiple of $\pi$
• C. an even multiple of $\frac{\pi }{2}$
• D. 0

Answer 1

Answer: (A)

an odd muliple of $\frac{\pi }{2}$

$A=\left[\begin{array}{cc}{\cos }^2\theta & \cos \theta \sin \theta \\ \cos \theta \sin \theta & {\sin }^2\theta \end{array}\right]B=\left[\begin{array}{cc}{\cos }^2\varphi & \cos \varphi \sin \varphi \\ \cos \varphi \sin \varphi & {\sin }^2\varphi \end{array}\right]AB=0\left[\begin{array}{cc}{\cos }^2\theta & \cos \theta \sin \theta \\ \cos \theta \sin \theta & {\sin }^2\theta \end{array}\right]\times \left[\begin{array}{cc}{\cos }^2\varphi & \cos \varphi \sin \varphi \\ \cos \varphi \sin \varphi & {\sin }^2\varphi \end{array}\right]=0\left[\begin{array}{cc}{\cos }^2\theta {\cos }^2\varphi +\cos \varphi \cos \theta \sin \theta \sin \varphi & {\cos }^2\theta \cos \varphi \sin \varphi +{\sin }^2\varphi \cos \theta \sin \theta \\ {\cos }^2\varphi \cos \theta \sin \theta +{\sin }^2\theta \cos \varphi \sin \varphi & \cos \theta \sin \theta \cos \varphi \sin \varphi +{\sin }^2\theta {\sin }^2\varphi \end{array}\right]=0\left[\begin{array}{cc}\cos \theta \cos \varphi (\cos \theta \cos \varphi +\sin \theta \sin \varphi )& \cos \theta \sin \varphi (\cos \theta \cos \varphi +\sin \varphi \sin \theta )\\ \sin \theta \cos \varphi (\cos \varphi \cos \theta +\sin \theta \sin \varphi )& \sin \theta \sin \varphi (\cos \theta \cos \varphi +\sin \theta \sin \varphi )\end{array}\right]=0(\cos \theta \cos \varphi +\sin \theta \sin \varphi )\left[\begin{array}{cc}\cos \theta \cos \varphi & \cos \theta \sin \varphi \\ \sin \theta \cos \varphi & \sin \theta \sin \varphi \end{array}\right]=0\cos \theta \cos \varphi +\sin \theta \sin \varphi =0\cos (\theta -\varphi )=0$

$\theta -\varphi =(2n+1)\frac{\pi }{2}$ where $n=0,1,.......$

$\theta -\varphi$ is an odd multiple of $\frac{\pi }{2}$