Proof

We use the property of the transposed matrix :

$$(AB{)}^t=B^t{A}^t$$

Then we have

$$\begin{array}{rcl}{A}^t(A^{-1}{)}^t& =& (A^{-1}A{)}^t=I\\ (A^{-1}{)}^t{A}^t& =& (A{A}^{-1}{)}^t=I\end{array}$$

where $I$ is an identity matrix.

This shows that the inverse of $A^t$ is the transpose of $A^{-1}$ :

$$(A^t{)}^{-1}=(A^{-1}{)}^t$$