(Solved):   2. If points on the \( \mathrm{xy} \)-plane are represented by \( 2 \ ...

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2. If points on the \( \mathrm{xy} \)-plane are represented by \( 2 \mathrm{x} 1 \) column matrices \( \left[\begin{array}{l}x \\ y\end{array}\right] \) then the matrix \( R(\theta) \), defined below, rotates points clockwise in an angle \( \theta \) in radians. \[ R(\theta)=\left[\begin{array}{cc} \cos (\theta) & \sin (\theta) \\ -\sin (\theta) & \cos (\theta) \end{array}\right] \] With this in mind, which of the following statements are true? 1) \( R(0)=I \) 2) \( R(\alpha) R(\beta)=R(\alpha+\beta) \) 3) \( R(\alpha) R(\beta)=R(\alpha \beta) \) 4) \( R^{-1}(\alpha)=R(-\alpha) \) 5) \( R^{-1}(\alpha)=R(1 / \alpha) \) 6) \( R^{n}(\alpha)=R(n \alpha) \) (A) Only 3) is False (B) Only 3), 5) and 6) are False (C) They are all True (D) Only 2) is False (E) Only 3) and 5) are False