Let $A$ be a unital commutative Banach algebra with maximal ideal space $X.$ We determine the rational H-type of the group $GL_n (A)$ of invertible n by n matrices with coefficients in A, in terms of the rational cohomology of $X.$ We also address an old problem of J. L. Taylor. We express the rational cohomology groups of the space of maximal ideals of A in terms of a topological invariant associated to A. Joint work with Greg Lupton, Chris Phillips and Claude Schochet cf. preprint posted to the arXiv