Determinants of Orthogonal Matrices

Theorem: Determinant of an Orthogonal Matrix

The Determinant of an Orthogonal Matrix is either $+1$ or $-1$.

PROOF

From $A^{\mathsf{T}}A=I_n$ we have

$$\det (A^{\mathsf{T}}A)=\det (I_n)$$ $$\det (A^{\mathsf{T}})\det (A)=1$$ $$\det (A{)}^2=1⟹\det (A)=\pm 1$$