I am trying to solve the following problem:
Given is a $3\times 3$ matrix $M$ over $\mathbb{F}_{7}$, such that for every vectors $v,w\in \mathbb{F}_{7}^3\setminus \{0\}$ there exists an integer $n$ with $M^{n}v=w$. Find this $M$.
Well, i think i have to look for matrices with full ranks, but i have no idea how and where to start... I've been thinking of idempotent matrices, but the problem is what to do with the vectors $v,w$. I can't just choose them to be (1,1,1).
Does anyone have an idea how can this probem be solved? I will be glad to read your comments and remarks. Thank you in advance!
Let $g$ be a primitive element of the field $E=\Bbb{F}_{7^3}$, and let $M$ represent multiplication by $g$ on $E$ viewed as a 3-dimensional vector space over $\Bbb{F}_7$. Show that this $M$ works.