  When K is an arbitrary field, we study the affine automorphisms of M_n(K)
that stabilize GL_n(K). Using a theorem of Dieudonn\'e on maximal affine
subspaces of singular matrices, this is easily reduced to the known case of
linear preservers when n>2 or #K>2. We include a short new proof of the more
general Flanders' theorem for affine subspaces of M_{p,q}(K) with bounded rank.
We also find that the group of affine transformations of M_2(F_2) that
stabilize GL_2(F_2) does not consist solely of linear maps. Using the theory of
quadratic forms over F_2, we construct explicit isomorphisms between it, the
symplectic group Sp_4(F_2) and the symmetric group S_6.
