Consider the matrix f(x)=cosxsinx0−sinxcosx0001.
Given below are two statements :
Statement I : f(−x) is the inverse of the matrix f(x).
Statement II : f(x)f(y)=f(x+y).
In the light of the above statements, choose the correct answer from the options given below :
Solutions · 1
Official solution
\[\begin{aligned}
& f(-x)=\left[\[\begin{array}{|ccccccc|}\cos x & \sin x & 0 \\ -\sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\end{aligned}\]\right] \\
\& f(x) \cdot f(-x)=\left[\[100010001\]\right]=I
\]
Hence statement- I is correct
Now, checking statement II
\[\begin{aligned}
& f(y)=\left[\[\begin{array}{|ccccccc|}\cos y & -\sin y & 0 \\ \sin y & \cos y & 0 \\ 0 & 0 & 1\end{array}\end{aligned}\]\right] \\
\& f(x) \cdot f(y)=\left[\[cos(x+y)sin(x+y)0−sin(x+y)cos(x+y)0001\]\right] \\
\& \Rightarrow f(x) \cdot f(y)=f(x+y)
\]
Hence statement-II is also correct.