Question

If $ A= \begin{bmatrix}cos^{2} \,\alpha&cos\, \alpha \,sin \,\alpha\\ cos\,\alpha\, sin \,\alpha&sin^{2} \,\alpha\end{bmatrix} $ and $ B= \begin{bmatrix}cos^{2}\, \beta &cos\, \beta \, sin\, \beta \\ cos\,\beta \,sin \,\beta &sin^{2} \,\beta \end{bmatrix} $ are two matrices such that the product $ AB $ is null matrix, then $ \alpha -\beta $ is

• A. $ 0 $
• B. multiple of $ \pi $
• C. an odd multiple of $ \pi/2 $
• D. None of the above

Answer 1

Answer: (C)

Given, $AB=O$ $\therefore \begin{bmatrix}cos^{2} \alpha&cos\,\alpha\,sin\,\alpha\\ cos\,\alpha\,sin\,\alpha&sin^{2}\,\alpha\end{bmatrix}$
$\times\begin{bmatrix}cos^{2}\,\beta&cos\,\beta\,sin\,\beta\\ cos\,\beta\,sin\,\beta&sin^{2} \beta\end{bmatrix}=\begin{bmatrix}0&0\\ 0&0\end{bmatrix}$
$\begin{bmatrix}cos\,\alpha\,cos\,\beta\,cos\left(\alpha-\beta\right)&cos\,\alpha\,sin\,\beta\,cos\left(\alpha-\beta\right)\\ cos\,\beta\,sin\,\alpha\,cos \left(\alpha-\beta\right)&sin\,\alpha\,sin\,\beta\,cos\left(\alpha-\beta\right)\end{bmatrix}$
$=\begin{bmatrix}0&0\\ 0&0\end{bmatrix}$
$\Rightarrow cos\left(\alpha-\beta\right)=0 $
$\Rightarrow \alpha-\beta$ is an odd multiple of $\pi /2 $