Product of Proper Orthogonal Matrices is Proper Orthogonal Matrix

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

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

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

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

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

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

From Determinant of Matrix Product:


 * $\map \det {\mathbf {P Q} } = \map \det {\mathbf P} \map \det {\mathbf Q}$

Thus:


 * $\map \det {\mathbf {P Q} } = 1$

Hence the result.