Determinantal representation and subschemes of general plane curves

Let M = (m_{ij}) be an nxn square matrix of integers. For our purposes, we can assume without loss of generality that M is homogeneous and that the entries are non-increasing going leftward and downward. Let d be the sum of the entries on either diagonal. We give a complete characterization of which such matrices have the property that a general form of degree d in C[x_0,x_1,x_2] can be written as the determinant of a matrix of forms (f_{ij}) with deg f_{ij} = m_{ij} (of course f_{ij} = 0 if m_{ij} < 0). As a consequence, we answer the related question of which (n-1)xn matrices Q of integers have the property that a general plane curve of degree d contains a zero-dimensional subscheme whose degree Hilbert-Burch matrix is Q. This leads to an algorithmic method to determine properties of linear series contained in general plane curves.