Matrices Question 3
Question 3 - 2024 (27 Jan Shift 1)
Consider the matrix $f(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \ \sin x & \cos x & 0 \ 0 & 0 & 1\end{array}\right]$.
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
(1) Statement I is false but Statement II is true
(2) Both Statement I and Statement II are false
(3) Statement I is true but Statement II is false
(4) Both Statement I and Statement II are true
Show Answer
Answer (4)
Solution
$f(-x)=\left[\begin{array}{ccc}\cos x & \sin x & 0 \ -\sin x & \cos x & 0 \ 0 & 0 & 1\end{array}\right]$
$f(x) \cdot f(-x)=\left[\begin{array}{lll}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}\right]=I$
Hence statement- I is correct
Now, checking statement II
$f(y)=\left[\begin{array}{ccc}\cos y & -\sin y & 0 \ \sin y & \cos y & 0 \ 0 & 0 & 1\end{array}\right]$
$f(x) \cdot f(y)=\left[\begin{array}{ccc}\cos (x+y) & -\sin (x+y) & 0 \ \sin (x+y) & \cos (x+y) & 0 \ 0 & 0 & 1\end{array}\right]$
$\Rightarrow \mathrm{f}(\mathrm{x}) \cdot \mathrm{f}(\mathrm{y})=\mathrm{f}(\mathrm{x}+\mathrm{y})$
Hence statement-II is also correct.
