Unit Matrix is Orthogonal

Theorem

The unit matrix $\mathbf I_n$ of order $n$ is orthogonal.

Proof

By Unit Matrix is its own Inverse the inverse $I_n^{-1}$ of $I_n$ is $I_n$.

By definition a unit matrix is a diagonal matrix.

Hence by Diagonal Matrix is Symmetric:

$I_n = I_n^\intercal$

where $I_n^\intercal$ is the transpose of $I_n$.

Thus:

$I_n^{-1} = I_n^\intercal$

and the result follows by definition of orthogonal.

$\blacksquare$