elementary matrix

When an elementary matrix multiplies a square matrix from the left, it applies to that matrix an elementary row operation homologous to the one that created its own self. (Likewise, when an elementary matrix multiplies a square matrix from the right, it applies to that matrix an elementary column operation homologous to the one that created its own self.)

If a series of elementary matrices applied to a given square matrix row-reduce it to the identity matrix, then the same series of elementary matrices applied to the identity matrix yield the inverse of the given matrix.