Compute $\left[M M^{T}\right]^{-1}$ for an orthogonal matrix where
\[M=\left[\begin{array}{lll}
\frac{1}{\sqrt{2}} & \frac{2}{\sqrt{2}} & \frac{-2}{\sqrt{2}} \\
\frac{-2}{\sqrt{2}} & \frac{2}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
\frac{2}{\sqrt{2}} & \frac{1}{\sqrt{2}} & \frac{2}{\sqrt{2}}
\end{array}\right] .\]