We fix a matrix $A \in \mathcal{M}_d(\mathbb{R})$, with characteristic polynomial $\chi_A(X) = \det(A - X \cdot I_d) = \sum_{k=0}^d a_k X^k$. Deduce that $\chi_A(A)$ is the zero matrix. One may use cofactor matrices.
We fix a matrix $A \in \mathcal{M}_d(\mathbb{R})$, with characteristic polynomial $\chi_A(X) = \det(A - X \cdot I_d) = \sum_{k=0}^d a_k X^k$.
Deduce that $\chi_A(A)$ is the zero matrix. One may use cofactor matrices.