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Question: If \(A = \begin{bmatrix} \cos\alpha & \sin\alpha \\ - \sin\alpha & \cos\alpha \end{bmatrix},\text{t...

If $A = \begin{bmatrix} \cos\alpha & \sin\alpha \

  • \sin\alpha & \cos\alpha \end{bmatrix},\text{then}A^{2} =$
A

[cos⁡2αsin⁡2αsin⁡2αcos⁡2α]\begin{bmatrix} \cos 2\alpha & \sin 2\alpha \\ \sin 2\alpha & \cos 2\alpha \end{bmatrix}

B

[cos⁡2α−sin⁡2αsin⁡2αcos⁡2α]\begin{bmatrix} \cos 2\alpha & - \sin 2\alpha \\ \sin 2\alpha & \cos 2\alpha \end{bmatrix}

C

$\begin{bmatrix} \cos 2\alpha & \sin 2\alpha \

  • \sin 2\alpha & \cos 2\alpha \end{bmatrix}$
D

$\begin{bmatrix}

  • \cos 2\alpha & \sin 2\alpha \
  • \sin 2\alpha & - \cos 2\alpha \end{bmatrix}$
Answer

$\begin{bmatrix} \cos 2\alpha & \sin 2\alpha \

  • \sin 2\alpha & \cos 2\alpha \end{bmatrix}$
Explanation

Solution

Since A2=A⋅A=[cos⁡αsin⁡α−sin⁡αcos⁡α][cos⁡αsin⁡α−sin⁡αcos⁡α]A ^ { 2 } = A \cdot A = \left[ \begin{array} { c c } \cos \alpha & \sin \alpha \\ - \sin \alpha & \cos \alpha \end{array} \right] \left[ \begin{array} { c c } \cos \alpha & \sin \alpha \\ - \sin \alpha & \cos \alpha \end{array} \right]

\cos 2\alpha & \sin 2\alpha \\ - \sin 2\alpha & \cos 2\alpha \end{bmatrix}$$