Prove that pre-multiplying a matrix A m by the elementary matrix obtained with any matrix elementary line transformation I m ⟶ l 1 ↔ l 2 E is the same as applying said elementary line transformation on the matrix A m

Name: Prove that pre-multiplying a matrix A m by the elementary matrix obtained with any matrix elementary line transformation I m ⟶ l 1 ↔ l 2 E is the same as applying said elementary line transformation on the matrix A m
Uploaded: 2022-02-23T17:22:57+00:00
Description: Prove that pre-multiplying a matrix A m by the elementary matrix obtained with any matrix elementary line transformation I m ⟶ l 1 ↔ l 2 E is the same as applying said elementary line transformation on the matrix A m

Question

Prove that pre-multiplying a matrix       A    m   by the elementary matrix obtained with any matrix elementary line transformation       I    m        ⟶                  l        1            ↔              l        2              E is the same as applying said elementary line transformation on the matrix       A    m

Alexzander Bowman · Accepted Answer

Let       e    1    ,  …  ,      e    n   be the standard basis row vectors, and observe the following facts about matrix multiplication:1.       e    i    A gives the ith row of the matrix   A2. For row vectors       v    1    ,  …  ,      v    m   , we have      (                                                   v            1                                                           ⋮                                                  v            m                                )    A  =      (                                                   v            1                    A                                                 ⋮                                                  v            m                    A                      )  Combine these to get the desired result for any elementary row operations.