  Let $M = (m_{ij})$ be an $n \times n$ 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
$\mathbb C[x_0,x_1,x_2]$ 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 $(n-1) \times n$
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.
