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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
Statement $I$ is false but Statement $II$ is true
Both Statement $I$ and Statement $II$ are false
Statement $I$ is true but Statement $II$ is false
Both Statement $I$ and Statement $II$ are true
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 f(x) \cdot f(y)=f(x+y)$
Hence statement -$II$ is also correct.
Similar Questions
Consider the following information regarding the number of men and women workers in three factories $I,\,II$ and $III$
| Men workers |
Women workers |
|
| $I$ | $30$ | $25$ |
| $II$ | $25$ | $31$ |
| $III$ | $27$ | $26$ |
Represent the above information in the form of a $3 \times 2$ matrix. What does the entry in the third row and second column represent?
