  We study the anti-commuting variety which consists of pairs of anti-commuting
$n\times n$ matrices. We provide an explicit description of its irreducible
components and their dimensions. The GIT quotient of the anti-commuting variety
with respect to the conjugation action of $GL_n$ is shown to be of pure
dimension $n$. We also show the semi-nilpotent anti-commuting variety (in which
one matrix is required to be nilpotent) is of pure dimension $n^2$ and describe
its irreducible components.
