Find the characteristic of the following field $F$.

Question

Could someone please verify whether my solution is fine?

Consider the following field:

$F=\left \{ \begin{bmatrix} 1& 0\\ 0& 1 \end{bmatrix},\begin{bmatrix} 1 & 1\\ 1& 0 \end{bmatrix},\begin{bmatrix} 0 &1 \\ 1 & 1 \end{bmatrix},\begin{bmatrix} 0&0 \\ 0 &0 \end{bmatrix} \right \}$

with entries from $\Bbb{Z}_{2}$. Find the characteristic of $F$.

Since $I\in F$ and for any $A\in F$, $AI=IA=A$, then $F$ has identity $I$.

Since $2\begin{bmatrix} 1 &0 \\ 0 & 1 \end{bmatrix}=\begin{bmatrix} 2&0 \\ 0 & 2 \end{bmatrix}=\begin{bmatrix} 0 &0 \\ 0 & 0 \end{bmatrix}$ (because $2\equiv 0$ (mod $2$)), then $\operatorname{char}(F)=|I|=2$.

Answer 1

Yes, it is correct. But it would have been faster to observe that, for any $M\in F$, $M+M=0$.