  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.
