QJEEM24Ma2d7f: JEE Main 2024 Mathematics - Class 12 JEEM24Ma2d7f Solution: Consider the matrix . Given below are two statements :…

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

Hence statement- I is correct

Now, checking statement II

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

Hence statement-II is also correct.