Which of the following statements about the adjugate (adjoint) matrix $\text{adj}(A)$ are true for any invertible $n \times n$ matrix $A$?

I. $A \cdot \text{adj}(A) = \det(A) \cdot I$

II. $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$

III. $\det(\text{adj}(A)) = (\det(A))^{n-1}$