Inverse of Orthogonal Matrix is Orthogonal

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

Then its inverse $\mathbf A^{-1}$ is also orthogonal.

Proof
By definition of orthogonal matrix:
 * $\mathbf A^\intercal = \mathbf A^{-1}$

where $\mathbf A^\intercal$ is the transpose of $\mathbf A$.

By Inverse of Inverse of Matrix:
 * $\left({\mathbf A^{-1} }\right)^{-1} = \mathbf A$

By Transpose of Transpose of Matrix:
 * $\left({\mathbf A^\intercal}\right)^\intercal = \mathbf A$

Thus we have:
 * $\left({\mathbf A^{-1} }\right)^{-1} = \mathbf A = \left({\mathbf A^\intercal}\right)^\intercal = \left({\mathbf A^{-1}}\right)^\intercal$

and so by definition $\mathbf A^{-1}$ is orthogonal.