I want to calculate the number of $2 \times 2$ matrices over the finite field $\mathbb{F}_q$ whose minimal polynomial is divisible by $X-1$.
The characteristic polynomial must be $(X-1)(X-a)$ for some $a \in \mathbb{F}_q$. If $a \neq 1$ then each of them is similar to $\text{diag}(1,a)$. but how do I count the number of matrices in this case ?
Another case if $a=1$, then either the matrix is $2 \times 2 $ identity or similar to its JCF which is upper triangular with all nonzero entry $1$. But again how do I count total number of matrices in each cases except the identity situation ?
I need help.
Best Answer
Expanding on Greg Martin's comment...
Let's find the number of matrices $\big( {s\atop u}{t\atop v} \big)$ with $(s-1)(v-1)=tu$.
Case 1; $t\ne 0$
$u$ is uniquely determined by $u=\frac{(s-1)(v-1)}{t}$. There are $(q-1)q^2$ ways to choose $s,v,t$ so that $t\ne 0$.
Case 2; $t= 0$
If $s=1, v\ne 1$ then there are $(q-1)q$ ways to choose $v$ and $u$. Likewise if $v=1, s\ne 1$. With $s=v=1$ there are $q$ ways to choose $u$.
So the total number of matrices with eigenvalue $1$ is $(q-1)q^2+2(q-1)q+q=q^3+q^2-q$.