Let $A$ be a square matrix in characteristic $p > 0$ with both column and row having length $(1 + p^0 + p + \cdots + p^i)$, where $i \geq 0$.

Suppose that further the $(m,n)$-component $a_{m,n}$ of the matrix $A$ is defined as follows$\colon$
Now, we shall give the presentation of $\sigma$ in terms of the matrix in the following manner$\colon$
\begin{align}\label{MATRIX}
&
a_{1,1} = a_{2,1} = \cdots = a_{1 + p^0 + \cdots + p^k + 1,1} = 1 \phantom{A} {\mathrm{for}} \phantom{A} 0 \leq k < i \notag \\
&
a_{2,1} = a_{4, 3} = \cdots = a_{m, m-1} = 1 \phantom{A} {\mathrm{for}} \phantom{A} m \not= 1 + p^0 + p + \cdots + p^k + 1, {\mathrm{where}}~0 \leq k < i \notag \\
&
a_{1 + p^0 + \cdots + p^k + 1,1 + p^0 + p + \cdots + p^k + p^{k + 1}} = 1 \phantom{A} {\mathrm{for}} \phantom{A} 0 \leq k < i, \notag \\
&
a_{m,n} = 0 \phantom{A} {\mathrm{otherwise.}}
\end{align}
where $a_{m,n}$ is the $(m,n)$-component of the matrix. For example in the case of $p = 2$ and $i = 2$, it is written as follows$\colon$

\begin{pmatrix}\label{matrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0
\end{pmatrix}

## Q. We shall consider the column vector $v$ of length $1 + p^0 + p^1 + \cdots + p^i$ and the condition $(A^e - I)v = 0$. Suppose the first entry of $v$ is $not$ zero. Then the minimal possible such $e$ must be $p^{i+1}$. Moreover, $A^{p^{i + 1}} = I$ must hold.