I learned reading this question that GL(n,p) elements have at most a multiplicative order of $p^n -1$.

I would like to know how many matrices have an order of exactly $p^n -1$. Do they represent a majority of matrices in GL(n,p) ?

The answer suggested to build matrices of maximal order this way.

Consider a degree n monic polynomial $P_n$ whose root is a generator $ξ$ of $F^∗_{p^n}$. Then a matrix with $P_n$ as its characteristic polynomial has order at least $p^n−1$ since $ξ$ is its eigenvalue.

I don't know how to create a such matrix, yet enumerate them...

Thank you so much for any help ^^

I learned reading this question that $\mathrm{GL}(n,p)$ elements have at most a multiplicative order of $p^n -1$.

I would like to know how many matrices have an order of exactly $p^n -1$. Do they represent a majority of matrices in $\mathrm{GL}(n,p)$?

The answer suggested to build matrices of maximal order this way.

Consider a degree n monic polynomial $P_n$ whose root is a generator $ξ$ of $F^∗_{p^n}$. Then a matrix with $P_n$ as its characteristic polynomial has order at least $p^n−1$ since $ξ$ is its eigenvalue.

I don't know how to create a such matrix, yet enumerate them...

Thank you so much for any help.