Example of a representable matroid

Question

I'm studying theorem 6.6.3, Oxley. It states that a matroid $M$ is representable over every field if and only if $M$ is binary and for some field $K$ of characteristic other than 2, $M$ is representable over $K$. I'd like to exhibit an example of a matroid that has an explicit representation over $F_2$ and over another field and then conclude that it's representable over every field using this theorem. Any suggestions for me?

Answer 1

Check out the examples on the wikipedia page for totally unimodular matrices. The connection (which I think is also mentioned in Oxley) is that a matroid is representable over every field if and only if it is representable by a totally unimodular matrix.

For a small concrete example consider the signed incidence matrix of some arbitrary orientation of the complete graph $K_3$: $$\begin{pmatrix}1&1&0\\-1&0&1\\0&-1&-1\end{pmatrix}.$$