How to prove that the following matrices in $M_p(\Bbb F_p)$ is similar:
Consider two matrices $$(a_{ij})= \begin{pmatrix} 1 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 1 & \cdots & 0 & 0\\ 0 & 0 & 1 & \cdots & 0 & 0\\ \vdots & \vdots & \vdots& \ddots & \vdots &\vdots\\ 0 & 0 & 0 & \cdots & 1 & 1\\ 0 & 0 & 0 & \cdots & 0 & 1\\ \end{pmatrix}$$ and $$(b_{ij})= \begin{pmatrix} 0 & 0 & 0 & \cdots & 0 & 1 \\ 1 & 0 & 0 & \cdots & 0 & 0\\ 0 & 1 & 0 & \cdots & 0 & 0\\ \vdots & \vdots & \vdots& \vdots & \vdots &\vdots\\ 0 & 0 & 0 & \cdots & 1 & 0\\ \end{pmatrix}$$
I can observe that the characteristic polynomial of $(a_{ij})$=minimal polynomial of $(a_{ij})$=$(x-1)^p$ and the characteristic polynomial of $(b_{ij})=x^p-1=(x-1)^p$
Now how do I argue from here that the minimal polynomial of $b_{ij}$ is also $(x-1)^p$?
$A,B$ are similar because $\dim(\ker(A-I))=1$ ($A$ is a Jordan block) and $\dim(\ker(B-I))=1$ ($B$ is a Frobenius block).