Let $A \in M_n(F[x])$ be an invertible matrix. Prove that $\deg(\det A)=0$.

Question

Let $A \in M_n(F[x])$ be an invertible matrix. Prove that $\deg(\det A)=0$

I succeeded proving it using $\det(AB)=\det A\det B$, without using the adjoint matrix like I was instructed to do.

I need your help proving it using that $A\cdot\mathrm{Adj}(A)=(\det A)I$.

Thank you!