When reducing a matrix using Gauss Jordan reduction, why are the corresponding elementary matrices acting on rows multipled together in reverse order?

Question

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?

Answer 1

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 $$ E_n(\cdots(E_2(E_1 M))\cdots) = (E_n \cdots E_2 E_1) M. $$