Let $R$ be a unital ring with multiplicative identity $1_R$, let $n \ge 1$, and let $A \in GL_n(R)$. For each integer $m \ge 0$, let $I_m \in GL_m(R)$ denote the $m \times m$ identity matrix over $R$, with $I_0$ interpreted as the empty identity block so that $\operatorname{diag}(A,I_0)=A$. Regard $A$ as an element of the stable general linear group $GL(R)=\varinjlim_k GL_k(R)$ under the stabilization maps $B \mapsto \operatorname{diag}(B,1_R)$. Let $E(R) \trianglelefteq GL(R)$ be the stable elementary subgroup generated by all elementary transvections $e_{ij}(r)=I_k+rE_{ij}$, where $k \ge 2$, $1 \le i \ne j \le k$, $r \in R$, and $E_{ij} \in M_k(R)$ is the standard matrix unit whose $(i,j)$ entry is $1_R$ and whose other entries are $0_R$. Let $[A]_{K_1}$ denote the class of $A$ in $K_1(R)=GL(R)/E(R)$.

Then, for every $m \ge 0$, $[\operatorname{diag}(A,I_m)]_{K_1}=[A]_{K_1}$. Moreover, for every $m \ge 0$ and every elementary matrix $e=e_{ij}(r) \in GL_{n+m}(R)$, where $1 \le i \ne j \le n+m$ and $r \in R$, one has $[e\,\operatorname{diag}(A,I_m)]_{K_1}=[\operatorname{diag}(A,I_m)]_{K_1}$ and $[\operatorname{diag}(A,I_m)e]_{K_1}=[\operatorname{diag}(A,I_m)]_{K_1}$. Equivalently, the class of $A$ in $K_1(R)$ is invariant under stable elementary row and column operations.

Consequently, if $R=\mathbb Z[G]$ for a group $G$, then the image of $A$ in the Whitehead group $\operatorname{Wh}(G)=K_1(\mathbb Z[G])/\langle [\pm g] : g \in G\rangle$ is invariant under stable elementary row and column operations, with the only additional identifications being those generated by the classes of the units $\pm g$.