Question

Let A =$\begin{bmatrix}cos^{2}\,\theta&sin\,\theta cos\,\theta \\ cos\,\theta sin\,\theta &sin^{2}\,\theta \end{bmatrix}$ and $B = \begin{bmatrix}cos^{2}\,\phi &sin\,\phi cos\,\phi\\ cos\,\phi sin\,\phi&sin^{2}\,\phi\end{bmatrix} $,then $AB = 0$,if

• A. $\theta =n\phi, n = 0, 1, 2, .... $
• B. $\theta +\phi= n\pi, n = 0, 1, 2, .... $
• C. $\theta =\phi+(2n + 1 ) \frac{\pi}{2}, n = 0, 1, 2, .... $
• D. $\theta =\phi+ \frac{n\pi}{2}, n = 0, 1, 2, .... $

Answer 1

Answer: (C)

AB = $\begin{bmatrix}cos^2\,\theta&sin\, \theta cos\,\theta\\ cos\,\theta sin\,\theta&sin^2\,\theta\end{bmatrix}\begin{bmatrix}cos^2\,\phi&sin\,\phi cos\,\phi\\ cos\phi sin\,\phi&sin^2\,\phi\end{bmatrix}$
.$=\begin{bmatrix}cos^{2}\,\theta cos^{2}\,\phi+sin\,\theta cos\,\phi cos\,\theta sin\,\phi&cos^{2}\,\theta sin\,\phi cos\,\phi +sin^{2}\,\phi sin\,\theta cos\,\theta\\ cos^{2}\, \phi cos\,\theta sin\,\theta+sin^{2}\,\theta sin\,\phi cos\,\phi&cos\, \theta sin\, \theta sin\,\phi cos\,\phi +sin^{2}\,\theta sin^{2}\,\phi\end{bmatrix}$
$=\begin{bmatrix}cos\,\theta cos\,\phi cos\left(\theta-\,\phi\right)&sin\phi cos\,\theta cos\left(\theta+\phi\right)\\ sin\theta cos\phi cos\left(\theta+\phi\right)&sin\theta sin\phi cos\left(\theta-\phi\right)\end{bmatrix}$
$\therefore AB = 0$
$\Rightarrow cos(\theta-\phi) = 0 $
$\Rightarrow cos(\theta-\phi) = cos(2n + 1 )\frac{\pi}{2}$
$\Rightarrow \theta = (2n + 1 )\frac{\pi}{2} + \phi$ ,where $n = 0, 1, 2, ......$