Determinant of Orthogonal Matrix is Plus or Minus One

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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:

\(\ds \mathbf Q \mathbf Q^\intercal\) \(=\) \(\ds \mathbf I\) Product of Orthogonal Matrix with Transpose is Identity
\(\ds \leadsto \ \ \) \(\ds \map \det {\mathbf Q \mathbf Q^\intercal}\) \(=\) \(\ds \det \mathbf I\)
\(\ds \leadsto \ \ \) \(\ds \map \det {\mathbf Q \mathbf Q^\intercal}\) \(=\) \(\ds 1\) Determinant of Unit Matrix
\(\ds \leadsto \ \ \) \(\ds \det \mathbf Q \det \mathbf Q^\intercal\) \(=\) \(\ds 1\) Determinant of Matrix Product

Hence the result.

$\blacksquare$


Sources