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Let $f: \mathbb R^3\to \mathbb R^3$ be the linear mapping which reflects $x$ over the plane $x1+x2+x3=0$. You are given that the standard matrix for $f$ is:

$$[f]=\frac13\left[\begin{array}{ccc}1&-2&-2\\-2&1&-2\\-2&-2&1\end{array}\right]$$

without using the row reduction technique for finding inverses, determine $[f]^{-1}$ and explain how you obtained your answer

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  • $\begingroup$ Here's a tutorial on how to format your questions. $\endgroup$ – user137731 Oct 26 '14 at 15:56
  • $\begingroup$ Think about this geometrically, how could you "undo" a reflection? $\endgroup$ – user137731 Oct 26 '14 at 15:57
  • $\begingroup$ Ahh, by reflecting it back again! $\endgroup$ – SJW Oct 26 '14 at 16:35
  • $\begingroup$ Yep. Good job! :) $\endgroup$ – user137731 Oct 26 '14 at 16:39
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By definition, a reflection is its own inverse. Sure enough, it turns out $[f]\circ [f] = I$, and therefore $[f]^{-1}=[f]$.

No row reduction as required :)

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  • $\begingroup$ Thank you so much! It all makes sense geometrically too :) $\endgroup$ – SJW Oct 26 '14 at 16:35

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