Question

If $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],J=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]\mathrm{and}B=\left[\begin{array}{cc}\cos \mathrm{\theta }& \sin \mathrm{\theta }\\ -\sin \mathrm{\theta }& \cos \mathrm{\theta }\end{array}\right]$, then B equals
(a) I cos θ + J sin θ
(b) I sin θ + J cos θ
(c) I cos θ − J sin θ
(d) −I cos θ + J sin θ

Answer 1

$$\begin{gathered} \left(a\right)I\cos \theta +J\sin \theta \\ \mathrm{Here}, \\ I\cos \theta +J\sin \theta \\ =\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\cos \theta +\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]\sin \theta \\ =\left[\begin{array}{cc}\cos \theta & 0\\ 0& \cos \theta \end{array}\right]+\left[\begin{array}{cc}0& \sin \theta \\ -\sin \theta & 0\end{array}\right] \\ =\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]=B \end{gathered}$$