  Let $A$ be a unital $C^*$-algebra and let $U_0(A)$ be the group of unitaries
of $A$ which are path connected to the identity. 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. It is also injective
when $A$ is a unital $C^*$-algebra of real rank zero, or $A$ has no tracial
state. We prove that the map is an isomorphism when $A$ is the Villadsen's
simple AH--algebra of stable rank $k>1.$ We also prove that the map is an
isomorphism for all Blackadar's unital projectionless separable simple
$C^*$-algebras. Let $A=\mathrm M_n(C(X)),$ where $X$ is any compact metric
space. It is noted that the map $i_A^{(1, n)}$ is an isomorphism for all $n.$
As a consequence, the map $i_A^{(1, n)}$ is always an isomorphism for any
unital $C^*$-algebra $A$ that is an inductive limit of finite direct sum of
$C^*$-algebras of the form $\mathrm M_n(C(X))$ as above. Nevertheless we show
that there are unital $C^*$-algebras $A$ such that $i_A^{(1,2)}$ is not an
isomorphism.
