The elementary matrices generate the general linear group GLn(F) when F is a field.
There are three types of elementary matrices, which correspond to three types of row operations (respectively, column operations): If E is an elementary matrix, as described below, to apply the elementary row operation to a matrix A, one multiplies A by the elementary matrix on the left, EA.
This fact can be understood as an instance of the Yoneda lemma applied to the category of matrices.
The corresponding elementary matrix is a diagonal matrix, with diagonal entries 1 everywhere except in the ith position, where it is m. So Di(m)A is the matrix produced from A by multiplying row i by m. Coefficient wise, the Di(m) matrix is defined by : The final type of row operation on a matrix A adds row j multiplied by a scalar m to row i.
And A Lij(m) is the matrix produced from A by adding m times column i to column j. Coefficient wise, the matrix Li,j(m) is defined by :