Product of Orthogonal Matrices is Orthogonal Matrix

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

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

Then $\mathbf P \mathbf Q$ is an orthogonal matrix.

Proof
From Determinant of Orthogonal Matrix is Plus or Minus One and Matrix is Invertible iff Determinant has Multiplicative Inverse it follows that both $\mathbf P$ and $\mathbf Q$ are invertible.

Thus:

Hence the result, by definition of orthogonal matrix.