Question

If $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],B=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$ and $C=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$, then $C=$

• A. $I\cos \theta +B\sin \theta$
• B. $I\sin \theta +B\cos \theta$
• C. $I\cos \theta -B\sin \theta$
• D. $-I\cos \theta +B\sin \theta$

Answer 1

The correct option is A $I\cos \theta +B\sin \theta$

Given: $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right];B=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$
$I\cos \theta =\left[\begin{array}{cc}\cos \theta & 0\\ 0& \cos \theta \end{array}\right]$
Similarly, we have, $B\sin \theta =\left[\begin{array}{cc}0& \sin \theta \\ -\sin \theta & 0\end{array}\right]$
On observation, we find, $I\cos \theta +B\sin \theta =\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]=C$