Inverse of Proper Orthogonal Matrix is Proper Orthogonal

Theorem
Let $\mathbf A$ be a proper orthogonal matrix.

Then its inverse $\mathbf A^{-1}$ is also proper orthogonal.

Proof
By definition of proper orthogonal matrix:
 * $\mathbf A$ is orthogonal
 * the determinant of $\mathbf A$ is equal to $1$.

By Inverse of Orthogonal Matrix is Orthogonal, $\mathbf A^{-1}$ is orthogonal.

By Determinant of Inverse Matrix:
 * $\det \mathbf A^{-1} = \dfrac 1 {\det \mathbf A} = 1$

Hence, by definition, $\mathbf A^{-1}$ is proper orthogonal.