We have the matrix $$M=\begin{pmatrix}4 & 1 & 1 \\ 1 & 1 & 1 \\ 2 & 1 & -1\end{pmatrix}$$ I want to find the lower triangular matrix $E$ of the elimination.
Is this matrix $E$ the same as the matrix $L$ at the $LU$ decomposotion?
Or how do we caclulate the matrix $E$ ?
I am interpreting that you want to find $E$ such that $EM=R$ where $R$ is a row echelon form.
As you perform elementary row operations to set it to a row echelon form without row swapping and you use your pivot entry to set those entries below the pivot entry to zero, your elementary matrix would be a lower triangular matrices, multiplying these matrices together would give you $E$.
If you have $M=LU$, then we have $$L^{-1}M=U$$
You can set $E=L^{-1}$.