Let $A$ be the vertex-edge incidence matrix of a graph, and let $A’$ be the matrix obtained from $A$ by adding an arbitrary row. The matroid $M[A’]$ is known as an even-cycle matroid. It is an instance of the class of lift matroids frequently discussed on this blog, such as by Irene ( here , here , and here ) and two weeks ago by Daryl ( here ). Note that it can be obtained from $M$ by coextending the matroid by one element, and then deleting that element. They can be visualized by coloring the edges of the graph, calling an edge even if its corresponding column in the matrix has a 0 in the new row, and odd if it has a 1. This class of matroids is closed under minors.