grandes-ecoles 2024 Q17

grandes-ecoles · France · x-ens-maths__psi Matrices Matrix Algebra and Product Properties

Let $B \in \mathbb{M}_n(u)$.
(a) Show that there exists a polynomial $R \in \mathbb{C}[X]$ such that $$u(A) = R(A) \text{ and } u(B) = R(B).$$ (b) We assume that $AB \in \mathbb{M}_n(u)$ and $BA \in \mathbb{M}_n(u)$. Show that $$A\, u(BA) = u(AB)\, A$$

Let $B \in \mathbb{M}_n(u)$.\\
(a) Show that there exists a polynomial $R \in \mathbb{C}[X]$ such that
$$u(A) = R(A) \text{ and } u(B) = R(B).$$
(b) We assume that $AB \in \mathbb{M}_n(u)$ and $BA \in \mathbb{M}_n(u)$. Show that
$$A\, u(BA) = u(AB)\, A$$

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