Let \(F\left(x\right)=\left[\begin{array}{lll}\cos x & -\sin x& 0\\ \sin x& \cos x& 0\\ 0& 0 &1\end{array}\right]\). Which of the following relation is satisfied by \(F\left(x\right)\).

\(F\left(x\right)F\left(y\right)=F\left(x+y\right)\)

The correct answer is Option (3) → \(F\left(x\right)F\left(y\right)=F\left(x+y\right)\)

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

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

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

$∴F(x)F(y)=F(x+y)$