Given $A$ is a nonsingular $3\times3$ matrix prove that if $AA^T=I$ then it is not necessary that $A^TA=I.$ Ie show shat for a matrix to be orthogonal its necessary that both $AA^T=I$ and $A^TA=I.$
Attempts- i tried to use variables for A but could not do it. As a habit if i take A^-1 =AT then the solution becomes wrong.
Since $AA^T=I$ you can check that $\det(A)\neq 0$ and therefore $A$ has an inverse $B$ (this means $BA=AB=I$). But we then have $$A^T=A^TI=A^T(AB)=(A^TA)B=B$$ and since $B=A^T$ is an inverse for $A$ we have that $A^TA=I$. This is a consequence of a more general statement regarding left and right inverses of invertible elements.
