Question

If $A_{\theta }$ = $\left(\begin{array}{cc}cos\theta & sin\theta \\ -sin\theta & cos\theta \end{array}\right)$ then $A_{\alpha }.A_{\beta }=$

• A. $A_{a+\beta }$
• B. $A_{a\beta }$
• C. $A_a+A_{\beta }$
• D. 1

Answer 1

Answer: (A)

$A_{a+\beta }$

$A_{\alpha }=\left(\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right)$
$A_{\beta }=\left(\begin{array}{cc}\cos \beta & \sin \beta \\ -\sin \beta & \cos \beta \end{array}\right)$
$A_{\alpha }\cdot A_{\beta }=\left(\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right)$
$\left(\begin{array}{cc}\cos \beta & \sin \beta \\ -\sin \beta & \cos \beta \end{array}\right)$
$=\left(\begin{array}{cc}\cos \alpha \cos \beta -\sin \alpha \sin \beta & \cos \alpha \sin \beta +\sin \alpha \cos \beta \\ -\sin \alpha \cos \beta -\sin \beta \cos \alpha & -\sin \alpha \sin \beta +\cos \alpha \cos \beta \end{array}\right)$
$=\left(\begin{array}{cc}\cos (\alpha +\beta )& \sin (\alpha +\beta )\\ -\sin (\alpha +\beta )& \cos (\alpha +\beta )\end{array}\right)$
$=A_{\alpha +\beta }$