Let $A$ be an invertible $n\times n$ matrix and let $B= \operatorname{adj}(A)$ be the classical adjoint of $A$.
I want to show that the characteristic polynomials $\phi_A(t)$ and $\phi_B(t)$ of $A$ and $B$, respectively satisfy
\begin{align}(-t)^n \phi_A\left( \frac{\det(A)}{t} \right) = \det(A) \phi_B(t). \end{align}
I know $A \operatorname{adj}(A) = \det(A) I_n$, and $\det(c A) = c^n \det(A)$, but having problem dealing with characteristic polynomials