Does every invertible matrix A has a matrix B such that A=Adj(B)? I'm trying to understand if it's always true, always true over $\mathbb C$ or never true.
I know that if $A$ is invertible, than there exists $A^{-1}$.
$$A=\frac{1}{det (A^{-1})}Adj(A^{-1})$$
So I have an adjoint matrix multiplied by a scalar, but how do I know if the result is an adjoint by itself?

There is no solution over $\mathbb R$ if $n \ge 3$ is odd and $\det(A) < 0$.
$\det(\text{adj}(B))= \det(\det(B) B^{-1}) = \det(B)^{n-1}$, which can't be negative in this case.
