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"](https://pdfs.semanticscholar.org/cdc1/a502e91748f0a55158e41880c82c35ba5eac.pdf) - which is not easy to read. Is there an easier argument?



2x2-determinantal representations of cubic curves
Let $(\ell\_1,\ell\_2)$ be the ideal of a point in $V(f)$, so $f \in (\ell\_1,\ell\_2)$.


