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Denote by $CU(A)$ the closure of the commutator subgroup of $U_0(A).$ Let $i_A^{(1, n)}\\colon U_0(A)/CU(A)\\rightarrow U_0(\\mathrm M_n(A))/CU(\\mathrm M_n(A))$ be the \\hm\\, defined by sending $u$ to ${\\rm diag}(u,1_n).$ We study the problem when the map $i_A^{(1,n)}$ is an isomorphism for all $n.$ We show that it is always surjective and is injective when $A$ has stable rank one. 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