Let $A \in M_p(\mathbb{C})$. Let $k \in \mathbb{N}^*$ and $\lambda_1, \lambda_2, \ldots, \lambda_k$ be the roots of $\chi_A$ pairwise distinct, with respective multiplicities $n_1, n_2, \ldots, n_k$. For every integer $q$ between 1 and $p$, $J_q$ denotes the matrix of $M_q(\mathbb{C})$ whose coefficients are all zero except those just above the diagonal which equal 1. Let $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ be the block diagonal matrix defined by $$B = \begin{pmatrix} \lambda_1 I_{n_1} + J_{n_1} & 0 & \ldots & 0 \\ 0 & \lambda_2 I_{n_2} + J_{n_2} & \cdots & 0 \\ \vdots & \ddots & \ddots & 0 \\ 0 & \ldots & 0 & \lambda_k I_{n_k} + J_{n_k} \end{pmatrix}$$ Show that $\chi_B = \chi_A$.
Let $A \in M_p(\mathbb{C})$. Let $k \in \mathbb{N}^*$ and $\lambda_1, \lambda_2, \ldots, \lambda_k$ be the roots of $\chi_A$ pairwise distinct, with respective multiplicities $n_1, n_2, \ldots, n_k$. For every integer $q$ between 1 and $p$, $J_q$ denotes the matrix of $M_q(\mathbb{C})$ whose coefficients are all zero except those just above the diagonal which equal 1.
Let $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ be the block diagonal matrix defined by
$$B = \begin{pmatrix} \lambda_1 I_{n_1} + J_{n_1} & 0 & \ldots & 0 \\ 0 & \lambda_2 I_{n_2} + J_{n_2} & \cdots & 0 \\ \vdots & \ddots & \ddots & 0 \\ 0 & \ldots & 0 & \lambda_k I_{n_k} + J_{n_k} \end{pmatrix}$$
Show that $\chi_B = \chi_A$.