Proof for one of the properties of Orthogonal matrices

Question

Consider a matrix O, let's assume it has orthonormal basis. If this... $$o^{T}=o^{-1}$$ is satisfied, then 0 is a orthogonal matrix. But how does one go to prove that the inverse of an orthogonal matrix is equal to its transpose? (Basically can someone proof the equation above for me)

Answer 1

This is effectively the definition of an orthogonal matrix, but to see it take a simple example, let $\{o_1,o_2\}$ be some orthonormal basis for $\mathbb{R}^2$ and let $O=[o_1\;o_2]$. Then by definition of the orthonormal basis we know that $OO^{\top}=I_2=OO^{-1}$. Note that $O^{-1}$ is unique and exists due to $\mathrm{det}(O)=1$.