If $a \in S$ is some invertible element in a ring $S$, then a computation shows
$\pmatrix{a & 0 \\ 0 & a^{-1}} = \pmatrix{1 & a \\ 0 & 1} \pmatrix{1 & 0 \\ -a^{-1} & 1} \pmatrix{1 & a \\ 0 & 1} \pmatrix{0 & -1 \\ 1 & 0}.$
If $R \to S$ is a surjective homomorphism, we see that this invertible matrix over $S$ may be lifted to some invertible matrix over $R$. This observation is important in algebraic K-Theory; for example it is used in the exactness of the relative $K_0$-sequence (see Rosenberg's book, 1.5.4 - 1.5.5).
Questions. What is the idea behind this factorization? Of course it makes no problem to verify this identity, but how can you come up with such a nontrivial factorization? Does it have a geometric interpretation? Who was the first one to find and use this identity?
PS: Isn't it sad that only few textbooks and papers offer explanations of the important insights, rather than only proof verifications?