A matrix of linear forms which is annihilated by a vector of indeterminates

Let R be a standard graded polynomial ring in f variables over a field and Psi be an f by g matrix of linear forms from R, where g is positive and less than f. Assume that the row vector of variables annihilates Psi and that the ideal I generated by the g by g minors of Psi has grade exactly one short of the maximum possible grade. We resolve R/I, prove that I has a g-linear resolution, record explicit formulas for the h-vector and multiplicity of R/I, and prove that if f-g is even, then the ideal I is unmixed. Furthermore, if f-g is odd, then we identify an explicit generating set for the unmixed part, I^{unm}, of I, resolve R/I^{unm}, and record explicit formulas for the h-vector of R/I^{unm}. These results have applications to the study of the blow-up algebras associated to linearly presented grade three Gorenstein ideals.