# On the exponent of a certain matrix $A$ in characteristic $p > 0$

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$$\colon$$

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