  We investigate the variety of commuting matrices. We classify its components
for any number of matrices of size at most 7. We prove that starting from
quadruples of size 8 matrices, this scheme has generically nonreduced
components, while up to degree 7 it is generically reduced. Our approach is to
recast the problem as deformations of modules and generalize an array of
methods: apolarity, duality and Bia\l{}ynicki-Birula decompositions to this
setup. We include a thorough review of our methods to make the paper
self-contained and accessible to both algebraic and linear-algebraic
communities. Our results give the corresponding statements for the Quot schemes
of points, in particular we classify the components of
$Quot_d(O_{\mathbb{A}^n}^{\oplus r})$ for $d\leq 7$ and all $r$, $n$.
