Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$. Prove that, if $A$ is a nilpotent matrix of index $p$, then every polynomial in $\mathbb{C}[X]$ that is a multiple of $X^p$ is an annihilating polynomial of $A$.
Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$.
Prove that, if $A$ is a nilpotent matrix of index $p$, then every polynomial in $\mathbb{C}[X]$ that is a multiple of $X^p$ is an annihilating polynomial of $A$.