How do I prove or show this? Not sure how to go about it.
If you can prove that both: $\displaystyle A^T \left(A^{-1}\right)^T = \left(A^{-1}\right)^T A^T = I$
then by definition, we've shown that $\displaystyle A^T$ is invertible.
So for: $\displaystyle A^T \left(A^{-1}\right)^T = I$
Use the fact that: $\displaystyle X^TY^T = (YX)^T$
So we see that: $\displaystyle A^T\left(A^{-1}\right)^T = \left(A^{-1}A\right)^T = I^T = I$
Now finish off by proving the other.