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I have a hard time explaining this problem so I think I'll let this youtube link do the talking.

https://youtu.be/-voH_B21eXc?t=1177

In this link, you could see that he's trying to reduce a matrix to echelon form. However, he mentions that to get P, he has to multiply the reduction matrices from right to left order. I understand that matrix multiplication isn't commutative, so how does that factor into why the order multiplication is reversed from the order of row operations?

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  • $\begingroup$ They aren't reversed. Think about function composition: when we write $f\circ g\circ h$, we do $h$ first, $g$ second, and $f$ last. Same thing here. $\endgroup$ Commented Feb 16, 2022 at 22:55
  • $\begingroup$ What is the connection between each row operation and function composition? $\endgroup$
    – Mezzoforte
    Commented Feb 17, 2022 at 0:53
  • $\begingroup$ The elementary matrices are like functions actung on the matrix. The effect of the product is to perform the elementary row operation. So you can think of the product as the application of a function on the matrix... because that is exactly what it is. $\endgroup$ Commented Feb 17, 2022 at 2:09

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If $E$ is the elementary matrix associated with a row operation, then we apply this operation to a matrix $M$ by computing the product $EM$. If we want to apply two row operations with associated matrices $E_1,E_2$ to a matrix $M$, we apply the first operation by computing $E_1M$ and apply the second operation to this result by computing $$ E_2(E_1 M) = (E_2E_1)M. $$ Concordantly, applying $n$ operations with associated matrices $E_1,\dots,E_n$ is the same as computing the product $$ E_n(\cdots(E_2(E_1 M))\cdots) = (E_n \cdots E_2 E_1) M. $$

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