# 3x3 Singular Matrices of Linear Forms

## TL;DR

This paper classifies all 3x3 matrices of linear forms with zero determinant into four fundamental types, providing a clear geometric and algebraic understanding of their structure.

## Contribution

It identifies and explicitly describes the four irreducible components of the space of 3x3 singular matrices of linear forms.

## Key findings

- Four irreducible components identified and described.
- Matrices are equivalent to one of four canonical forms under elementary operations.
- Provides a concrete classification of 3x3 singular matrices of linear forms.

## Abstract

We determine the irreducible components of the space of 3x3 matrices of linear forms with vanishing determinant. We show that there are four irreducible components and we identify them concretely. In particular, under elementary row and column operations with constant coefficients, a 3x3 matrix with vanishing determinant is equivalent to one of the following four: a matrix with a zero row, a zero column, a zero 2x2 square or an antisymmetric matrix.