Let A A be a Hopf algebra and ⌫ be a bicovariant first order differential calculus over A A. It is known that there are three possibilities to construct a differential Hopf algebra ⌫ n s ⌫ m rJ that contains ⌫ as its first order part. Corresponding to the three choices of the ideal J, we distinguish the ''universal'' exterior algebra, the ''second antisymmetrizer'' exterior algebra, and Woronowicz' external algebra, respectively. Let ⌫ be one of the N 2 -dimensional bicovariant first order differen-Ž . Ž . tial calculi on the quantum group GL N or SL N , and let q be a transcendenq q tal complex number. For Woronowicz' external algebra we determine the dimension of the space of left-invariant and of bi-invariant k-forms, respectively. Bi-invariant forms are closed and represent different de Rham cohomology classes. The algebra of bi-invariant forms is graded anti-commutative. For N G 3 the three Ž . differential Hopf algebras coincide. However, in case of the 4 D -calculi on SL 2 " q the universal differential Hopf algebra is strictly larger than Woronowicz' external algebra. The bi-invariant 1-form is not closed.