[tex]\underbrace{\begin{bmatrix}-2&-1&2\\-2&2&3\\-4&1&3\end{bmatrix}}_A\underbrace{\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}}_x=\underbrace{\begin{bmatrix}-1\\-1\\4\end{bmatrix}}_b[/tex]
Multiply [tex]A[/tex] on the left side with the following elimination matrix [tex]E_1[/tex]:
[tex]\underbrace{\begin{bmatrix}1&0&0\\-1&1&0\\-2&0&1\end{bmatrix}}_{E_1}A=\begin{bmatrix}-2&-1&2\\0&3&1\\0&3&-1\end{bmatrix}[/tex]
Multiply [tex]E_1A[/tex] on the left by another elimination matrix [tex]E_2[/tex]:
[tex]\underbrace{\begin{bmatrix}1&0&0\\0&1&0\\0&-1&1\end{bmatrix}}_{E_2}(E_1A)=\begin{bmatrix}-2&-1&2\\0&3&1\\0&0&-2\end{bmatrix}[/tex]
[tex]\implies\boxed{U=\begin{bmatrix}-2&-1&2\\0&3&1\\0&0&-2\end{bmatrix}}[/tex]
Multiply on the left by the inverse of [tex]E_2E_1[/tex]:
[tex](E_2E_1)^{-1}(E_2E_1)A=(E_2E_1)^{-1}U[/tex]
[tex]A=\underbrace{({E_1}^{-1}{E_2}^{-1})}_LU[/tex]
We have
[tex]{E_1}^{-1}=\begin{bmatrix}1&0&0\\1&1&0\\2&0&1\end{bmatrix}[/tex]
[tex]{E_2}^{-1}=\begin{bmatrix}1&0&0\\0&1&0\\0&1&1\end{bmatrix}[/tex]
[tex]\implies\boxed{L=\begin{bmatrix}1&0&0\\1&1&0\\3&1&1\end{bmatrix}}[/tex]
