Zero Determinant Rank 1 (2x2) matrix being expressed as an outer product of two commuting vectors (spinors)

Question

If a 2x2 matrix has a zero determinant, why can we express it as an (outer) product of two vectors? I'm working on the spinor-helicity formalism, and am curious as to the rigorous mathematical proof behind this. Any direction to literature would be very useful!

Thank you!

EDIT: See page 10 of https://arxiv.org/pdf/1308.1697.pdf for the kind of thing I'm interested in.

Answer 1

Thanks to @StubbornAtom's response, I found the answer that I was needing. Specifically relating to my project;

Since a zero determinant of any $n$ x $n$ matrix implies that the rank must be less than $n$, the rank for a 2x2 matrix must be 0 (null matrix) or 1. As a standard exercise in linear algebra, we can show that any rank-1 matrix may be written as the outer product of two vectors, a well-documented result in textbooks.