Question

If $A\left(\alpha \right)=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]$, Then the matrix $A^2\left(\alpha \right)=$

• A. $A\left(2\alpha \right)$
• B. $A\left(\alpha \right)$
• C. $A\left(3\alpha \right)$
• D. $A\left(4\alpha \right)$

Answer 1

The correct option is A

$A\left(2\alpha \right)$

Finding the value given matrix:

Given, $A\left(\alpha \right)=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]$

Now,

$\begin{array}{rcl}{A}^2\left(\alpha \right)& =& \left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]\times \left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]\\ A^2\left(\alpha \right)& =& \left[\begin{array}{cc}{\cos }^2\alpha -{\sin }^2\alpha & 2\cos \alpha \sin \alpha \\ -2\cos \alpha \sin \alpha & {\cos }^2\alpha -{\sin }^2\alpha \end{array}\right]\\ A^2\left(\alpha \right)& =& \left[\begin{array}{cc}\cos 2\alpha & \sin 2\alpha \\ -\sin 2\alpha & \cos 2\alpha \end{array}\right]\\ {\mathbf{A}}^{\mathbf{2}}\mathbf{\left(}\mathbf{\alpha }\mathbf{\right)}& \mathbf{=}& \mathbf{A}\mathbf{\left(}\mathbf{2}\mathbf{\alpha }\mathbf{\right)}\end{array}$

Hence, option $\mathbf{\left(}\mathit{A}\mathbf{\right)}$ is correct.