UPSC NDA I 2017 Mathematics Solved Paper

Concept:The matrix $A$ represents a rotation by $\theta$.Multiplying rotation matrices adds the angles.So $A^3$ corresponds to rotation by $3\theta$.Explanation:First compute $A^2 = A \cdot A$.$A^2 = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}$.Multiply: top-left: $\cos^2\theta - \sin^2\theta = \cos2\theta$.Top-right: $\cos\theta\sin\theta + \sin\theta\cos\theta = 2\sin\theta\cos\theta = \sin2\theta$.Bottom-left: $-\sin\theta\cos\theta - \cos\theta\sin\theta = -2\sin\theta\cos\theta = -\sin2\theta$.Bottom-right: $\cos^2\theta - \sin^2\theta = \cos2\theta$.Thus $A^2 = \begin{bmatrix} \cos2\theta & \sin2\theta \\ -\sin2\theta & \cos2\theta \end{bmatrix}$.Now multiply $A^2$ by $A$ to get $A^3$.$A^3 = \begin{bmatrix} \cos2\theta & \sin2\theta \\ -\sin2\theta & \cos2\theta \end{bmatrix} \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}$.Compute each entry using standard matrix multiplication.Top-left: $\cos2\theta\cos\theta - \sin2\theta\sin\theta = \cos(2\theta+\theta) = \cos3\theta$.Top-right: $\cos2\theta\sin\theta + \sin2\theta\cos\theta = \sin(2\theta+\theta) = \sin3\theta$.Bottom-left: $-\sin2\theta\cos\theta - \cos2\theta\sin\theta = -(\sin2\theta\cos\theta + \cos2\theta\sin\theta) = -\sin(2\theta+\theta) = -\sin3\theta$.Bottom-right: $-\sin2\theta\sin\theta + \cos2\theta\cos\theta = \cos2\theta\cos\theta - \sin2\theta\sin\theta = \cos3\theta$.Therefore $A^3 = \begin{bmatrix} \cos3\theta & \sin3\theta \\ -\sin3\theta & \cos3\theta \end{bmatrix}$.Answer:$\begin{bmatrix} \cos3\theta & \sin3\theta \\ -\sin3\theta & \cos3\theta \end{bmatrix}$, which matches option A.