If $\text{rank}(A) = 4$, then $\text{rank}(\text{adj}(A)) = 4$ and $\det(\text{adj}(A)) = (\det(A))^3$.

If $\text{rank}(A) = 3$, then $\text{rank}(\text{adj}(A)) = 1$ and $\det(\text{adj}(A)) = 0$.

If $\text{rank}(A) \leq 2$, then $\text{adj}(A)$ is the zero matrix.

If $\det(A) = 0$, then $\det(\text{adj}(A)) = 0$.