Product of Proper Orthogonal Matrices is Proper Orthogonal Matrix

Theorem
Let $\mathbf P$ and $\mathbf Q$ be proper orthogonal matrices.

Let $\mathbf P \mathbf Q$ be the (conventional) matrix product of $\mathbf P$ and $\mathbf Q$.

Then $\mathbf P \mathbf Q$ is a proper orthogonal matrix.

Proof
By definition, $\mathbf P \mathbf Q$ is a proper orthogonal matrix iff it is an orthogonal matrix with a determinant of $1$.

From Product of Orthogonal Matrices is Orthogonal Matrix, $\mathbf P \mathbf Q$ is an orthogonal matrix.

By definition, $\mathbf P$ and $\mathbf Q$ both have a determinant of $1$.

From Determinant of Matrix Product:


 * $\det \left({\mathbf P \mathbf Q}\right) = \det \left({\mathbf P}\right) \det \left({\mathbf Q}\right)$

Thus:


 * $\det \left({\mathbf P \mathbf Q}\right) = 1$

Hence the result.