Determinant of an orthogonal matrix

Computation. From $Q^T Q=I$:
$$\det(Q^T Q) = \det I = 1.$$
By multiplicativity of the determinant and $\det(Q^T)=\det Q$:
$$\det(Q)\det(Q) = 1.$$
So $(\det Q)^2=1$, hence $\det Q = \pm1$.

Interpretation. An orthogonal matrix represents an isometry of $\mathbb{R}^n$. If $\det Q=1$, it is a rotation; if $\det Q=-1$, it is a reflection composed with a rotation.