Question

Assertion :If $F\left(\alpha \right)=\left[\begin{array}{ccc}cos\alpha & -sin\alpha & 0\\ sin\alpha & cos\alpha & 0\\ 0& 0& 1\end{array}\right]$, then $\left[F\right(\alpha ){]}^{-1}=F(-\alpha )$. Reason: For matrix $G\left(\beta \right)=\left[\begin{array}{ccc}cos\beta & 0& sin\beta \\ 0& 1& 0\\ -sin\beta & 0& cos\beta \end{array}\right]$, we have $\left[G\right(\beta ){]}^{-1}=G(-\beta )$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

$F{\left(\alpha \right)}^{-1}=\frac{adjF\left(\alpha \right)}{\left|F\left(\alpha \right)\right|}=\frac{\left[\begin{array}{ccc}cos\alpha & sin\alpha & 0\\ -sin\alpha & cos\alpha & 0\\ 0& 0& 1\end{array}\right]}{1}=\left[\begin{array}{ccc}cos\alpha & sin\alpha & 0\\ -sin\alpha & cos\alpha & 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}cos(-\alpha )& -sin(-\alpha )& 0\\ sin(-\alpha )& cos(-\alpha )& 0\\ 0& 0& 1\end{array}\right]=F(-\alpha )$

Now

$G{\left(\beta \right)}^{-1}=\frac{adjG\left(\beta \right)}{\left|G\left(\beta \right)\right|}=\frac{\left[\begin{array}{ccc}cos\beta & 0& -sin\beta \\ 0& 1& 0\\ sin\beta & 0& cos\beta \end{array}\right]}{1}=\left[\begin{array}{ccc}cos(-\beta )& 0& sin(-\beta )\\ 0& 1& 0\\ -sin(-\beta )& 0& cos(-\beta )\end{array}\right]=G(-\beta )$

So Statement $1$ & $2$ are individually true.

But statement $2$ is not the correct explanation because these answers are just question specific.