Minimal non-orientable matroids of rank three

Minimal non-orientable matroids have been investigated as a threshold between orientability and non-orientability. The Fano matroid and the MacLane matroid are well-known minimal non-orientable matroids of rank 3. A natural question is whether there exists a minimal non-orientable matroid of every rank $r$ with $m$ elements. In this paper, we give an answer to the question in rank 3 that for every $m\ge 7$, there exists a minimal non-orientable matroid of rank 3 with $m$ elements. To prove this statement, we construct two new infinite families of minimal non-orientable matroids of rank 3. Furthermore we also investigate representability of the two infinite families.