If matrix A is orthogonal (dot product of each pair of its columns is zero; also dot product of each pair of its rows is 0; dot product of each row or column with itself is 1), show that $A^T = A^{-1}$.
I'm stuck with this proof because I'm not sure how to theoretically calculate $A^TA$ and $AA^T$.
Let $\langle,\rangle$ a scalar product on $R^n$, $A$ is orthogonal if and only if for every $x,y\in R^n$, $\langle x,y\rangle =\langle A(x),A(y)\rangle$. This is equivalent to saying that $\langle AA^T(x),y\rangle=\langle x,y\rangle$. This is equivalent to $AA^T=Id$.