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

Answer 1

I cos θ + J sin θ

\[Here, \]

\[I \cos \theta + J \sin \theta\]

\[ = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\cos \theta + \begin{bmatrix}0 & 1 \\ - 1 & 0\end{bmatrix}\sin \theta\]

\[ = \begin{bmatrix}\cos \theta & 0 \\ 0 & \cos \theta\end{bmatrix} + \begin{bmatrix}0 & \sin \theta \\ - \sin \theta & 0\end{bmatrix}\]

\[ = \begin{bmatrix}\cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix} = B\]