Product of Proper Orthogonal Matrices is Proper Orthogonal Matrix

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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 if and only if 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.

$\blacksquare$