Cristopher Knox

Answered

2022-07-01

Prove that pre-multiplying a matrix ${A}_{m}$ by the elementary matrix obtained with any matrix elementary line transformation ${I}_{m}\underset{{l}_{1}\leftrightarrow {l}_{2}}{\u27f6}E$ is the same as applying said elementary line transformation on the matrix ${A}_{m}$

Answer & Explanation

Alexzander Bowman

Expert

2022-07-02Added 19 answers

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

$\left(\begin{array}{c}\text{}{v}_{1}\text{}\\ \vdots \\ {v}_{m}\end{array}\right)A=\left(\begin{array}{c}\text{}{v}_{1}A\text{}\\ \vdots \\ {v}_{m}A\end{array}\right)$

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

1. ${e}_{i}A$ gives the ith row of the matrix $A$

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

$\left(\begin{array}{c}\text{}{v}_{1}\text{}\\ \vdots \\ {v}_{m}\end{array}\right)A=\left(\begin{array}{c}\text{}{v}_{1}A\text{}\\ \vdots \\ {v}_{m}A\end{array}\right)$

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

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