Product of Orthogonal Matrix with Transpose is Identity

Theorem
Let $\mathbf Q$ be an orthogonal matrix.

Then:
 * $\mathbf Q \mathbf Q^\intercal = \mathbf I = \mathbf Q^\intercal \mathbf Q$

where:
 * $\mathbf Q^\intercal$ is the transpose of $\mathbf Q$
 * $\mathbf I$ is a unit (identity) matrix

Proof
By definition, an orthogonal matrix is one such that:
 * $\mathbf Q^\intercal = \mathbf Q^{-1}$

and so the result follows by definition of inverse.