Answer
The statement is correct.
Work Step by Step
Since $A$ is invertible, then
$ A^{-1} =1/|A| * adj A $
and
$ (A^{-1})^{-1} =1/|A^{-1}| * adj A^{-1} $
and then
$ A^{-1} = ( (A^{-1})^{-1})^{-1} =|A^{-1}| * (adj A^{-1})^{-1} $
We know that
$|A^{-1}|=1/|A|$ and $ A^{-1} =1/|A| * adj A $
Therefore,
$1/|A| * adj A =|A^{-1}| * (adj A^{-1})^{-1} =(1/|A|) * (adj A^{-1})^{-1} $
Thus
$ adj A = (adj A^{-1})^{-1} $
and then
$ (adj A)^{-1} = adj A^{-1} $
