Question

Assertion :Let $A\left(\theta \right)=\left[\begin{array}{cc}\cos \theta +\sin \theta & \sqrt{2}\sin \theta \\ -\sqrt{2}\sin \theta & \cos \theta -\sin \theta \end{array}\right]$

$A(\pi /3{)}^3=-I$

Reason: $A\left(\theta \right)A\left(\varphi \right)=A(\theta +\varphi )$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Given that $A\left(\theta \right)=\left[\begin{array}{cc}\cos \theta +\sin \theta & \sqrt{2}\sin \theta \\ -\sqrt{2}\sin \theta & \cos \theta -\sin \theta \end{array}\right]$

Let $A\left(\theta \right)A\left(\varphi \right)=\left[\begin{array}{cc}a& b\\ -b& c\end{array}\right]$

where $a=(\cos \theta +\sin \theta )(\cos \varphi +\sin \varphi )-2\sin \theta \sin \varphi$
$=(\cos \theta \cos \varphi -\sin \theta \sin \varphi )+(\sin \theta \cos \varphi +\cos \theta \sin \varphi )$
$=\cos (\theta +\varphi )+\sin (\theta +\varphi )$;
$b=\sqrt{2}\left[\sin \varphi \right(\cos \theta +\sin \theta )+\sin \theta (\sin \varphi +\cos \varphi \left)\right]$

and $c=-2\sin \theta \sin \varphi +(\cos \theta -\sin \theta )(\cos \varphi -\sin \varphi )$
$=\cos \theta \cos \varphi -\sin \theta \sin \varphi -(\sin \theta \cos \varphi +\cos \varphi \sin \theta )$
$=\cos (\theta +\varphi )-\sin (\theta +\varphi )$

Thus, $A\left(\theta \right)A\left(\varphi \right)=A(\theta +\varphi )$
$\Rightarrow A(\theta {)}^2=A(2\theta )$
$A(\theta {)}^3=A(2\theta \left)A\right(\theta )=A(3\theta )$
$\therefore A(\pi /3{)}^3=A(\pi )=-I$

Thus both the Assertion and Reason are correct and Reason is the correct explanation for Assertion.