Theorem (Norm-Preserving Matrix)

Let $\mathbf{A}$ be a $k\times k$ matrix of real elements. If $\mathbf{A}$ is orthogonal then it is Norm-preserving i.e. $$\lVert \mathbf{Ax} \rVert=\lVert \mathbf{x} \rVert \ \ \forall x\in\mathbb{R}^{k}$$

Theorem (Orthogonal Matrices have Determinant 1)

Let $\mathbf{A}$ be a $k\times k$ matrix of real elements. If $\mathbf{A}$ is orthogonal then $\det \mathbf{A}=1$