# conditions for LU on a $2 \times 2$ matrix

I have $$A= \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$$ I want to do LU factorization on it, so I need to find the elementary matrix $E$ such that $EA=U$ and thus $L=E^{-1}$. So I know I need the row operation $aR_2-cR_1\rightarrow R_2$, but how do I represent $E$ with this row operation? I know that if it were just $R_2-cR_1\rightarrow R_2$ the elementary matrix would be $$\begin{bmatrix} 1 & 0 \\ -c & 1 \\ \end{bmatrix}$$ How do I get the $a$ involved?

$aR_2-cR_1\rightarrow R_2$ is not an elementary operation, but the result of two consecutive row operations. First, we set $aR_2 \to R_2$, then, we set $R_2 - cR_1 \to R_2$. So, we would have the product of two elementary matrices, namely $$E_2 E_1 = \begin{bmatrix} 1&0\\ -c&1 \end{bmatrix} \begin{bmatrix} 1 & 0\\ 0 & a \end{bmatrix} = \begin{bmatrix} 1 & 0\\ -c & a \end{bmatrix}$$