Question

If $R\left(t\right) = \begin{bmatrix}\cos t&\sin t\\ -\sin t&\cos t\end{bmatrix}$, 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) = \begin{bmatrix}\cos s&\sin s\\ -\sin s& \cos s\end{bmatrix} \times\begin{bmatrix}\cos t&\sin t\\ -\sin t&\cos t\end{bmatrix} $
$= \begin{bmatrix}\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{bmatrix} $
$= \begin{bmatrix}\cos\left(s+t\right)&\sin\left(s+t\right)\\ -\sin\left(s+t\right)&\cos\left(s+t\right)\end{bmatrix}=R\left(s+t\right) $