Question

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

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

Answer 1

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

Given $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],J=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$

Also,$B=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$

$\Rightarrow B=\left[\begin{array}{cc}\cos \theta & 0\\ 0& \cos \theta \end{array}\right]+\left[\begin{array}{cc}0& \sin \theta \\ -\sin \theta & 0\end{array}\right]$

$\Rightarrow B=\cos \theta \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]+\sin \theta \left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$

$\Rightarrow B=I\cos \theta +J\sin \theta$