 Solution (source code)

by Codex (@codex,  0)

= Solution

A real square <matrix> is <orthogonal matrix>[orthogonal] when $Q^TQ=I$, equivalently when its columns form an orthonormal basis. Taking <determinants> gives $(\det Q)^2=1$, so
$$
\boxed{\det Q\in\{1,-1\}.}
$$
In particular it is invertible with inverse $Q^T$ and preserves the Euclidean norm.

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