Question

Solve:
$F\left(x\right)=\left[\begin{array}{ccc}\cos x& -\sin x& 0\\ \sin x& \cos x& 0\\ 0& 0& 1\end{array}\right]$. Show that $f\left(x\right)f\left(y\right)=f(x+y)$

Answer 1

$f\left(x\right)=\left[\begin{array}{ccc}\cos x& -\sin x& 0\\ \sin x& \cos x& 0\\ 0& 0& 1\end{array}\right]$

$f\left(y\right)=\left[\begin{array}{ccc}\cos y& -\sin y& 0\\ \sin y& \cos y& 0\\ 0& 0& 1\end{array}\right]$

$f\left(x\right)\times f\left(y\right)$

$=\left[\begin{array}{ccc}\cos x& -\sin x& 0\\ \sin x& \cos x& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos y& -\sin y& 0\\ \sin y& \cos y& 0\\ 0& 0& 1\end{array}\right]$

$=\left[\begin{array}{ccc}\cos x\cos y-\sin x\sin y& -\cos x\sin y-\sin x\cos y& 0\\ \sin x\cos y+\cos x\sin y& \cos x\cos y-\sin x\sin y& 0\\ 0& 0& 1\end{array}\right]$

$=\left[\begin{array}{ccc}\cos (x+y)& -\sin (x+y)& 0\\ \sin (x+y)& \cos (x+y)& 0\\ 0& 0& 1\end{array}\right]$

$=f(x+y)$

Hence proved.