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:

$\paren {\mathbf A^{-1} }^{-1} = \mathbf A$

By Transpose of Transpose of Matrix:

$\paren {\mathbf A^\intercal}^\intercal = \mathbf A$

Thus we have:

$\paren {\mathbf A^{-1} }^{-1} = \mathbf A = \paren {\mathbf A^\intercal}^\intercal = \paren {\mathbf A^{-1} }^\intercal$

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

$\blacksquare$