Question

If $R\left(t\right)=\left[\begin{array}{cc}\cos t& \sin t\\ -\sin t& \cos t\end{array}\right]$, then R(s) R(t) equals

• A. R (s + t)
• B. R (s - t)
• C. R(s) + R(t)
• D. None of these

Answer 1

Answer: (A)

$R\left(s\right)R\left(t\right)=\left[\begin{array}{cc}\cos s& \sin s\\ -\sin s& \cos s\end{array}\right]\times \left[\begin{array}{cc}\cos t& \sin t\\ -\sin t& \cos t\end{array}\right]$
$=\left[\begin{array}{cc}\cos s\cos t-\sin s\sin t& \cos s\sin t+\sin s\cos t\\ -\sin s\cos t-\cos s\sin t& -\sin s\sin t+\cos s\cos t\end{array}\right]$
$=\left[\begin{array}{cc}\cos \left(s+t\right)& \sin \left(s+t\right)\\ -\sin \left(s+t\right)& \cos \left(s+t\right)\end{array}\right]=R\left(s+t\right)$