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Question
If \[I = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}, J = \begin{bmatrix}0 & 1 \\ - 1 & 0\end{bmatrix} and B = \begin{bmatrix}\cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix}\] then B equals )
Options
I cos θ + J sin θ
I sin θ + J cos θ
I cos θ − J sin θ
−I cos θ + J sin θ
Solution
I cos θ + J sin θ
\[Here, \]
\[I \cos \theta + J \sin \theta\]
\[ = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\cos \theta + \begin{bmatrix}0 & 1 \\ - 1 & 0\end{bmatrix}\sin \theta\]
\[ = \begin{bmatrix}\cos \theta & 0 \\ 0 & \cos \theta\end{bmatrix} + \begin{bmatrix}0 & \sin \theta \\ - \sin \theta & 0\end{bmatrix}\]
\[ = \begin{bmatrix}\cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix} = B\]
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