# Need help to find the inverse of a mapping implicitly

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

• Here's a tutorial on how to format your questions. – user137731 Oct 26 '14 at 15:56
• Think about this geometrically, how could you "undo" a reflection? – user137731 Oct 26 '14 at 15:57
• Ahh, by reflecting it back again! – SJW Oct 26 '14 at 16:35
• Yep. Good job! :) – user137731 Oct 26 '14 at 16:39

By definition, a reflection is its own inverse. Sure enough, it turns out $[f]\circ [f] = I$, and therefore $[f]^{-1}=[f]$.