On matrix spaces with zero determinant

Let g be a linear space of n×n matrices of determinant zero over an infinite (or suitably large finite) field. It is proved that if the dimension of. L exceeds n 2−2n+2, then either L or its transpose has a common null vector. This extends a result due to Dieudonne and solves a recent research problem posed by S. Pierce in this journal. We also consider the problem of classifying all maximal matrix spaces with zero determinant, and offer some examples and observations.