Before revision:  Under (semi)-strong identification, where standard asymptotics are valid, this matrix is asymptotically equivalent to the usual Jacobian matrix. After re-scaling, it is thus asymptotically non-singular.
After revision:  Under (semi)-strong identification, where standard asymptotics are valid, this matrix is asymptotically equivalent to the usual Jacobian and, after re-scaling, it is thus asymptotically non-singular.