elementary matrix
The simplest nontrivial thing you can do to a matrix is add a multiple of one row to another. The matrix that performs this single move — the identity with one extra entry off the diagonal — is an elementary matrix. These are the atoms of Gaussian elimination, and in K-theory they play the role of the moves you are allowed to ignore: K_1 is precisely what survives after you quotient out the subgroup they generate.
For a ring R, indices i not equal to j, and a ring element r, the elementary matrix e_{ij}(r) is the matrix that agrees with the identity except for the entry r in position (i, j). These matrices satisfy clean relations: e_{ij}(r) e_{ij}(s) = e_{ij}(r + s) (so each fixed pair (i, j) gives a subgroup isomorphic to the additive group of R), and the Steinberg commutator relations describing [e_{ij}(r), e_{kl}(s)] depending on how the indices overlap. The subgroup of GL(n, R) generated by all e_{ij}(r) is the elementary group E(n, R), and its stable union is E(R).
Each e_{ij}(r) has determinant 1, so E(R) lies inside SL(R). Over a field, or any Euclidean domain, Gaussian elimination shows E(n, R) = SL(n, R), so the elementary matrices generate everything of determinant 1. Over more general rings this can fail, and the gap SL(R) / E(R) is part of what SK_1 measures. The elementary matrices are also the geometric origin of the Steinberg group, whose abstract presentation copies exactly these relations.
In GL(2, R), e_{12}(r) = [1, r; 0, 1] and e_{21}(s) = [1, 0; s, 1]. Their product e_{12}(r) e_{21}(s) = [1 + rs, r; s, 1] already has nontrivial entries everywhere, showing how complicated matrices arise from a few elementary moves.
Two elementary 2x2 matrices and their product.
Beware that algebraists' elementary matrices are only the transvections e_{ij}(r); the row-scaling and row-swap matrices of an undergraduate linear-algebra course are deliberately excluded, since scaling carries the determinant information that K_1 is designed to retain rather than discard.