Orthogonal Matrix Conditions

Question

I am a bit confused about a matrix being orthogonal. A square matrix $A$ is said to be orthogonal if $AA^T = I$, and also if $A^T = A^{-1}$.

Now I was looking through the internet and found a third condition too i.e.: $$AA^T = A^TA = I$$ I want to ask that which of the above mentioned condition has to be checked in order to find if a matrix is orthogonal or not? Can we use any one of the above mentioned condition or do we need to test all these three?

Also can the condition $AA^{-1}=I$ also be used to check if a matrix is orthogonal?

Answer 1

In fact, they are all equivalent. $$AA^\intercal=I\iff A^\intercal=A^{-1}\iff A^\intercal A=I.$$