I have a question about a step of a proof in Atiyah Macdonald. It's the proposition 2.4.
Let M be a finitely generate A-module, let a be an ideal of A, and let $ \phi $ be an A-module endomorphism of M such that $$
\phi \left( M \right) \subset aM
$$
Then $\phi$ satisfies an equation of the form
$$
\eqalign{
& \phi ^n + a_1 \phi ^{n - 1} + ... + a_n = 0 \cr
& \text{ where }\ \ a_i \in a \cr}
$$
I don't understand why the determinant annihilates each $x_i$ because I did not understand the step of the adjoint matrix.