Linear varieties and matroids with applications to the Cullis' determinant

Let $V$ be a vector space of rectangular $n\times k$ matrices annihilating the Cullis' determinant. We show that $\dim(V) \le (n-1)k$, extending Dieudonn{\'{e}}'s result on the dimension of vector spaces of square matrices annihilating the ordinary determinant. Furthermore, for certain values of $n$ and $k$, we explicitly describe such vector spaces of maximal dimension. Namely, we establish that if $k$ is odd, $n \ge k + 2$ and $\dim(V) = (n-1)k$, then $V$ is equal to the space of all $n\times k$ matrices $X$ such that alternating row sum of $X$ is equal to zero. Our proofs rely on the following observations from the matroid theory that have an independent interest. First, we provide a notion of matroid corresponding to a given linear variety. Second, we prove that if the linear variety is transformed by projections and restrictions, then the behaviour of the corresponding matroid is expressed in the terms of matroid contraction and restriction. Third, we establish that if $M$ is a matroid, $I^*$ its coindependent set $M|S$ and its restriction on a set $S$, then the union of $I^*\setminus S$ with every cobase of $M|S$ is coindependent set of $M$.