It is hard to formulate a question, but I want to ask about a reference/recipe for computing syzygies in general.
For example, on $\mathbb{P}^1_{(x:y)}$ there is an exact sequence
$0\longrightarrow \mathcal{O}_{\mathbb{P}^1}(-3)^2\xrightarrow{\quad A\quad} \mathcal{O}_{\mathbb{P}^!}(-2)^3\xrightarrow{\quad (x^2,xy,y^2)\quad} \mathcal{O}_{\mathbb{P}^1}\longrightarrow0$
I was told $A= \begin{bmatrix} y & 0 \\ -x & y \\ 0 & -x \end{bmatrix}$.
Is there a theory that helps determine $A$? How can a person see this so easily? What is the intuition behind such a choice of $A$?
(I could write out the matrix and compute its entries without too much difficulty, but there must be a systematic theory?)
If the points are distinct -- i.e., do not coïncide -- AND the determinant of their coördinates vanishes, you have a syzygy. Is that what you are seeking?