Cristopher Knox

2022-07-01

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

Alexzander Bowman

Expert

Let ${e}_{1},\dots ,{e}_{n}$ 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 $A$
2. For row vectors ${v}_{1},\dots ,{v}_{m}$ , we have

Combine these to get the desired result for any elementary row operations.

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