groups

**thomas_donald** · August 29th 2009, 05:15 PM

I need help on the following problem:
Let a be the rotation about the origin in the plane and let p be the reflection in the x-axis. Show that pap-1=a-1.
Thanks

**ynj** · August 29th 2009, 06:41 PM

Originally Posted by thomas_donald

I need help on the following problem:
Let a be the rotation about the origin in the plane and let p be the reflection in the x-axis. Show that pap-1=a-1.
Thanks

you may use matrix to explain it.
the rotation can be represented as $a=\left(\begin{array}{cc}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{array}\right )$
the reflection can be represented as $p=\left(\begin{array}{cc}1&0\\0&-1\end{array}\right)$
So $pa=\left(\begin{array}{cc}\cos\theta&-\sin\theta\\-\sin\theta&-\cos\theta\end{array}\right)$
$papa=(pa)^2=\left(\begin{array}{cc}1&0\\0&1\end{ar ray}\right)=I$
Thus $pap=a^{-1}\Rightarrow pap^{-1}=a^{-1}$ since $p=p^{-1}$

**thomas_donald** · August 30th 2009, 07:37 PM

Thank you very much for your help.

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