Question

If $A=\left[\begin{array}{cc}{\cos }^2\alpha & \cos \alpha \sin \alpha \\ \cos \alpha \sin \alpha & {\sin }^2\alpha \end{array}\right]$
and $B=\left[\begin{array}{cc}{\cos }^2\beta & \cos \beta \sin \beta \\ \cos \beta \sin \beta & {\sin }^2\beta \end{array}\right]$
are two matrices such that $AB$ is the null matrix, then

• A. $\alpha =\beta$
• B. $\cos (\alpha -\beta )=0$
• C. $\sin (\alpha -\beta )=0$
• D. none of these

Answer 1

The correct option is B $\cos (\alpha -\beta )=0$

Given $AB=O$
$\left[\begin{array}{cc}{\cos }^2\alpha & \cos \alpha \sin \alpha \\ \cos \alpha \sin \alpha & {\sin }^2\alpha \end{array}\right]\left[\begin{array}{cc}{\cos }^2\beta & \cos \beta \sin \beta \\ \cos \beta \sin \beta & {\sin }^2\beta \end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}\cos \alpha \cos \beta \cos (\alpha -\beta )& \cos \alpha \sin \beta \cos (\alpha -\beta )\\ \cos \beta \sin \alpha \cos (\alpha -\beta )& \sin \alpha \sin \beta \cos (\alpha -\beta )\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

$\Rightarrow \cos (\alpha -\beta )=0$