Summary: For any (unital) exchange ring \(R\) whose finitely generated projective modules satisfy the separative cancellation property \((A\oplus A\cong A\oplus B\cong B\oplus B\Longrightarrow A\cong B)\), 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 H.–X.Lin [Pac.J.Math.173, No. 2, 443–489 (1996; Zbl 0860.46039)], 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 This verifies, in the separative case, a conjecture of Shuang Zhang [unpublished].