What does the notation $M_2(F_2)$ mean

Question

I suppose my main confusion is what matrices $M_2(F_2)$ actually contains. Would it just be matrices with elements 0 and 1?

Answer 1

Yes, it's the set of matrix $2\times 2$ with coefficient in $\mathbb F_2=\mathbb Z/2\mathbb Z$. In other words, $$\mathcal M_2(\mathbb F_2)=\left\{\begin{pmatrix}a&b\\c&d\end{pmatrix}\mid a,b,c,d\in \mathbb F_2=\{0,1\}\right\}$$

Answer 2

$M_2(F_2)$ is the ring of $2\times 2$ matrices with coefficients in $F_2$. Note that $M_2(F_2)$ is not a subset of $M_2(\mathbb{R})$.