Is there a matrix which is not orthogonal but only has A transpose A equal to identity?

Question

Can there be a matrix $A$ with $A A^T = I$ and $A^T A \neq I$ ? If so give a 2x2 example of A

Answer 1

There are no such finite square matrices. The equation $AA^T=I$ implies that $A^T=A^{-1}$ and matrices commute with their inverse.

Answer 2

$$A A^T = I$$ $$A^{-1}A A^T = A^{-1}I=A^{-1}$$ $$IA^T =A^T = A^{-1}$$ $$A^TA = A^{-1}A=I$$ $$A^TA = AA^{T}=I$$