# greatest common divisor of the slots of $\operatorname{adj}(tI_n-A)$ and characteristic polynomial

I'm trying to show:

Let $A\in \mathcal{M}_n(\mathbb{F})$ a matrix with characteristic polynomial $p_A(t)$ and minimal poliynomial $q_A(t)$. Let $d_{n-1}(t)$ 'greatest common divisor' of the slots of $\operatorname{adj}(tI_n-A)$.

a) Show that $d_{n-1}$ divided to $p_A(t)$. If $q^*(t)=\frac{p_A(t)}{d_{n-1}(t)}$ show that $q^*(t)=0$ and $q_A(t)$ divided $q^*(t)=$.

b) If $q^*(t)=s(t)q_A(t)$ in $\mathbb{F}(t)$, show that $s(t)\equiv 1$.

I was trying to use the equivalency: $$B^{-1}=\frac{1}{\det B} \operatorname{adj} B$$ $$\det B (B)^{-1}=\operatorname{adj}B$$ with $B=tI-A$, but I have two problems: if $t$ is a eigenvalues of $A$, the matrix $tI-A$ is not invertible and I dont know if $(tI-A)^{-1}$ have a polynomials slots.