Here is the question I am trying to solve letter $(b)$ in it:
Let $A$ be the matrix $\begin{pmatrix} 1 & 0 & 0 & 1 & 1 & 0\\ 0 & 1 & 0 & 1 & 0 & 1\\ 0 & 0 & 1 & 0 & 1 & 1 \end{pmatrix}$ For $q$ in $\{2,3\},$ let $M_q[A]$ be the vector matroid of $A$ when $A$ is viewed over $GF(q),$ the field of $q$ elements. Show that:
$(a)$ The sets of circuits of $M_2[A]$ and $M_3[A]$ are different. $(b)$ $M_2[A]$ is graphic but $M_3[A]$ is not.
My thoughts are:
I know that a matroid that is isomorphic to the cycle matroid of a graph is called graphic. So, how can I find the cycle matroid of a graph in case of GF(2) and how can I negate the existence of such a cycle matroid in GF(3)?
Could someone give me some hints please?