Solution
= Solution
The <orthogonal reflection> in $x=y$ exchanges the first two coordinates and leaves the third unchanged. Its fixed subspace is the plane, while its normal $(1,-1,0)$ changes sign. Therefore
$$
\boxed{S=\begin{pmatrix}0&1&0\\1&0&0\\0&0&1\end{pmatrix}.}
$$
This <reflection matrix> has <determinant> $-1$, as expected.