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Matrix Representations
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Which of the following matrices represents a rotation of 45 degrees about the origin in the $xy$-plane?
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A
$$\begin{bmatrix} \cos 45° & -\sin 45° \\ \sin 45° & \cos 45° \end{bmatrix}$$
💡 Explanation:
The matrix $$\begin{bmatrix} \cos 45° & -\sin 45° \ \sin 45° & \cos 45° \end{bmatrix}$$ represents a rotation of 45 degrees about the origin in the $xy$-plane because it satisfies the following equation: $$\begin{bmatrix} x' \ y' \end{bmatrix} = \begin{bmatrix} \cos 45° & -\sin 45° \ \sin 45° & \cos 45° \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}$$, where $$(x', y')$$ are the coordinates of a point after the rotation and $$(x, y)$$ are the coordinates of the point before the rotation.