# 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

$\det \mathbf Q^\intercal = \det \mathbf Q$

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.

$\blacksquare$