We fix a matrix $A \in \mathcal{M}_d(\mathbb{R})$. We consider the characteristic polynomial
$$\chi_A(X) = \det\left(A - X \cdot I_d\right) = \sum_{k=0}^d a_k X^k$$
Show that for $R$ large enough:
$$\chi_A(A) = \frac{1}{2\pi} \int_0^{2\pi} \left(R\mathrm{e}^{\mathrm{i}\theta}\right) \chi_A\left(R\mathrm{e}^{\mathrm{i}\theta}\right) \left(R\mathrm{e}^{\mathrm{i}\theta} I_d - A\right)^{-1} \mathrm{~d}\theta$$