Why if the columns of a matrix are orthogonal, then $AA^T\neq A^TA=I$?

Why if the columns of a matrix are orthogonal, then $$AA^T\neq A^TA=I$$? I know this isn't true if the columns are orthonormal, but since $$A^TA=I$$, then $$A^T=A^{-1}$$, and if exists the left inverse, then exists the right inverse such that $$AA^{T}=I$$. Am I missing something?

• $A$ doesn't have to be square to have orthogonal columns. So $A^TA = I$ doesn't necessarily imply that $A$ is invertible. – Matthew Leingang Jun 12 at 14:29
• What if we deal only with square matrices? – Jonathan S. Jun 12 at 14:31
• With orthogonal columns, you only have $A^TA$ diagonal, so you don't have $A^TA=I$. – user10354138 Jun 12 at 14:41
• The problem arises as I saw this in a question: "Now, if you work out what it means for the columns of $𝐴$ to be orthogonal, then that comes out as $𝐴^𝑇𝐴=𝐼_𝑛$. And, the rows of $𝐴$ are orthogonal if and only if $𝐴𝐴^𝑇=𝐼_𝑛$" math.stackexchange.com/a/2250266/268627 – Jonathan S. Jun 12 at 14:49
• @JonathanS. I don't see how that conflicts with anything said in the comment section here. If we deal with square matrices, then $A^TA=I$ and $AA^T=I$ are equivalent, and neither can happen without the other. – Arthur Jun 12 at 15:00