  For any (unital) exchange ring $R$ whose finitely generated projective
modules satisfy the separative cancellation property ($A\oplus A\cong A\oplus
B\cong B\oplus B$ implies $A\cong B$), it is shown that all invertible square
matrices over $R$ can be diagonalized by elementary row and column operations.
Consequently, the natural homomorphism $GL_1(R) \to K_1(R)$ is surjective. In
combination with a result of Huaxin Lin, it follows that for any separative,
unital C*-algebra $A$ with real rank zero, the topological $K_1(A)$ is
naturally isomorphic to the unitary group $U(A)$ modulo the connected component
of the identity. This verifies, in the separative case, a conjecture of Shuang
Zhang.
