Determinant of Orthogonal Matrix is Plus or Minus One

Theorem
Let $\mathbf Q$ be an orthogonal matrix.

Then:
 * $\det \mathbf Q = \pm 1$

where $\det \mathbf Q$ is the determinant of $\mathbf Q$.

Proof
By Determinant of Transpose:
 * $\det \mathbf Q^\intercal = \det \mathbf Q$

Then:

Hence the result.