What are the elements of this ring? Where $\Bbb F_5$ is the finite field of $5$ elements. I'm not sure what this notation is getting at:
$$ \mathbb F_5 \left(\begin{bmatrix}1&2\\2&4\end{bmatrix}\right)$$
I assumed it was just
$$ \left\{\begin{bmatrix}1&2\\2&4\end{bmatrix},\begin{bmatrix}2&4\\4&3\end{bmatrix},\begin{bmatrix}3&1\\1&2\end{bmatrix},\begin{bmatrix}4&3\\3&1\end{bmatrix},\begin{bmatrix}0&0\\0&0\end{bmatrix}\right\}$$
These matrices are nilpotent so multiplying them only generates the $0$ matrix
 A: I don't think there is a single ironclad interpretation in this case.  Part of the problem is that we don't know if the entries of the matrix are supposed to be from $F_5$ or not, and we don't clearly see what $F_5()$ means.
When $\alpha$ is an element of a field $K$ and $F$ is a field contained in $K$, $F(\alpha)$ usually denotes the smallest subfield of $K$ containing $F$ and $\alpha$.  However, no matter what ring you think the coefficients come from, the matrix is not invertible (check the determinant) so it's certainly not an element of a field contained in $M_2(R)$ for any ring $R$.
One could also interpret it as "the smallest $F_5$ algebra of $M_2(F_5)$ containing that matrix" and you'd get the set you gave plus the scalar $2\times 2$ matrices in $M_2(F_5)$.  Another way to think about this is $F_5[X]/(X^2)$ since $X^2$ is the minimal polynomial for this matrix over $F_5$.
One could still interpret it as "the smallest $F_5$ algebra (without identity) of $M_2(F_5)$ containing that matrix" and it would be the set you got (which is a zero ring: $R^2=\{0\}$) but it would not be a unitary algebra or contain any copy of $F_5$ like we usually want with algebras.  Usually this would be denoted as $F_5[\alpha]$ (note the change to brackets.)
One could also try to suppose that the coefficients of the matrix come from a field other than $F_5$ so that maybe it isn't nilpotent, but that raises the question of "if you wanted a matrix of infinite mulitplicative order then why did you pick that matrix then instead of something easy like $\begin{bmatrix}1&0\\0&2\end{bmatrix}$ in $M_2(\mathbb Z)$?"
If I had to guess, I'd pick the one in the third paragraph.
