Row-reducing the quantum determinant and Dieudonne determinant

We prove that row reducing a quantum matrix yields another quantum matrix for the same parameter q. This means that the elements of the new matrix satisfy the same relations as those of the original quantum matrix ring M_q(n). As a corollary, we can prove that the image of the quantum determinant in the abelianization of the total ring of quotients of M_q(n) is equal to the Dieudonne determinant of the quantum matrix. A similar result is proved for the multiparameter quantum determinant.