The relation between the minimal polynomial of $\textbf{A}$ and the monic greatest common divisor of all elements of $adj((s\textbf{I}-\textbf{A}))$

Question

From the relation of following equation

${(s\textbf{I}-\textbf{A})^{-1}} = \frac{1}{\Delta(s)} \text{adj}((s\textbf{I}-\textbf{A}))$,

the textbook says $\psi(s)=\Delta(s)/m(s)$.

$\Delta(s)$ means $\det(s\textbf{I}-\textbf{A})$,

$\psi(s)$ means the minimal polynomial of $\textbf{A}$,

and $m(s)$ means the monic greatest common divisor of all elements of $\text{adj}((s\textbf{I}-\textbf{A}))$.

But I can't derive the relation above. How to prove $\psi(s)=\Delta(s)/m(s)$?