Question about Multiplying Matrices

Reducing a matrix through row operations is the same as multiplying the matrix by the elementary matrices corresponding to each row operation (the elementary matrix corresponding to a given row operation is the matrix you get by applying that row operation to the identity matrix). That is, if matrix A can be reduced to A' by row operations corresponding to elementary matrices d₁, d₂... d_n and B can be reduced to B' by row operations corresponding to elementary matrices e₁, e₂, ..., e[sun]m[/sub], then A'= d₁d₂...d_nA and B'= e₁e₂...e_mB so that A'B'= d₁d₂...d_nAe₁e₂...e_mB which is, generally, NOT the same as AB.