Let $R$ be a commutative unital ring. For each $n \ge 1$, let $GL_n(R)$ be the group of invertible $n \times n$ matrices over $R$, and let $s_n:GL_n(R)\to GL_{n+1}(R)$ be the stabilization homomorphism $A\mapsto \operatorname{diag}(A,1_R)$. Let $GL(R)=\varinjlim_n GL_n(R)$ be the stable general linear group. For $n\ge2$, distinct indices $1\le i\ne j\le n$, and $r\in R$, let $e_{ij}(r)=I_n+rE_{ij}\in GL_n(R)$ be the elementary transvection, and let $E(R)\le GL(R)$ be the stable elementary subgroup generated by the images of all such transvections under stabilization. Define $K_1(R)=GL(R)/E(R)$. Then the ordinary determinant homomorphisms $\det_n:GL_n(R)\to R^\times$ induce a well-defined [group homomorphism](/page/Group%20Homomorphism) $\det:K_1(R)\to R^\times$ satisfying $\det([A])=\det_n(A)$ for every $A\in GL_n(R)$.