Let $A$ be a ring. Then we have the group $$ GL(A) = \bigcup_{i =1}^\infty GL_n(A) $$ which we can abelianize to form the torsion group $K(A)$. In the case where $A$ is commutative, we have a homorphism to the group of units $$ \text{det} : K(A) \to A^\times $$ which has kernel $SK(A)$. I was learning about this stuff by looking at Milnor's article on Whitehead torsion and he makes the remark that $SK(A) = 1$ if and only if every determinant one matrix in $GL(A)$ is equivalent to the identity matrix via some number of elementary row operations.
I do not understand either direction of this equivalence - I guess I am missing some linear algebra observations. Could someone elaborate on this equivalence?