Given a $3 \times 3$ array $$ A =\begin{bmatrix} 1 & 2 &1 \\ 3 & 8 & 1 \\ 0 &4 &1 \end{bmatrix}, $$
My understanding is that that I subtract $3$ times row one from row two to eliminate entry $a_{21}$. Therefore $E_{21}$ should be $$ E_{21} = \begin{bmatrix} 1 & 1 & 1 \\ \mathbf{0} & 2 & -2 \\ 0 & 4 & 1\end{bmatrix} $$ I see in my notes that elimination by multiplication yields $$ E_{21} = \begin{bmatrix} 1 & 0 & 0 \\ \mathbf{-3} & 1 & 1 \\ 0 & 0 & 1\end{bmatrix} $$.
I would like to understand the intuition behind the entries in the multiplication matrix. I am able to follow how one would find the entries for $E_{21}$ using the subtraction method (I highlighted $0$ to illustrate that I understand where that comes from, but I am supposed to use the multiplication method and I do not understand how one reaches these entries. Specifically the highlighted $-3$. Can someone please explain to me how to use the multiplication method to eliminate entries in a matrix?