Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. We assume throughout this part that $A$ is diagonalizable in $\mathscr{M}_n(\mathbb{C})$ and we denote by $\lambda_1, \cdots, \lambda_\ell$ its eigenvalues with $\lambda_i \neq \lambda_j$ if $i \neq j$. Show that
$$\varphi_A(X) = (X - \lambda_1) \cdots (X - \lambda_\ell).$$