In the talk I will prove that for any measurable mapping into the space of matrices with positive determinant, there is a diffeomorphism whose derivative equals outside a set of measure less than . Using this fact I will prove that for any measurable mapping into the space of matrices with non-zero determinant (with no sign restriction), there is an almost everywhere approximately differentiable homeomorphism whose derivative equals almost everywhere. The talk is based on my joint work with P. Goldstein and Z. Grochulska.