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

Let R=k[T1,…,Tf]R=k[T1,…,Tf] be a standard graded polynomial ring over the field k and Ψ be an f×gf×g matrix of linear forms from R  , where 1≤gIg(Ψ), prove that R‾ has a gg-linear resolution, record explicit formulas for the h  -vector and multiplicity of R‾, and prove that if f−gf−g is even, then the ideal Ig(Ψ)Ig(Ψ) is unmixed. Furthermore, if f−gf−g is odd, then we identify an explicit generating set for the unmixed part, Ig(Ψ)unmIg(Ψ)unm, of Ig(Ψ)Ig(Ψ), resolve R/Ig(Ψ)unmR/Ig(Ψ)unm, and record explicit formulas for the h  -vector of R/Ig(Ψ)unmR/Ig(Ψ)unm. (The rings R/Ig(Ψ)R/Ig(Ψ) and R/Ig(Ψ)unmR/Ig(Ψ)unm automatically have the same multiplicity.) These results have applications to the study of the blow-up algebras associated to linearly presented grade three Gorenstein ideals.