TITLE: 2x2-determinantal representations of cubic curves
QUESTION [7 upvotes]: Let $H_d:=\mathbb{C}_d[x,y,z]$ denote the space of homogeneous degree $d$ polynomials in $x$, $y$, $z$ with complex coefficients. I'd like to show that every $f\in H_3$ can be represented as
$$
f=\det\begin{pmatrix}q_1 & q_2\\ \ell_1 & \ell_2\end{pmatrix}, q_i\in H_2, \ell_i\in H_1, i=1,2.
$$
This is different from the usual determinantal representations, where each matrix entry is a linear form.
I suspect that the answer is true, and it can be proved using technique
from A.Beauville's "Determinantal hypersurfaces" - which is not easy to read. Is there an easier argument?

REPLY [9 votes]: Let $(\ell_1,\ell_2)$ be the ideal of a point in $V(f)$, so $f \in (\ell_1,\ell_2)$.