So, in the answer I don't understand how they obtained

$\displaystyle L= \begin{pmatrix}1 & 0 & 0 \\ 3 & 1 & 0 \\ 1 & -1 & 1 \end{pmatrix}$

I don't think this "L" here is the lower triangular matrix used in the LU factorization of A. Because I followed the LU decomposition algorithm and ended up with

$\displaystyle L= \begin{pmatrix}1 & 0 & 0 \\ 3 & 8 & 0 \\ 1 & 4 & 5 \end{pmatrix}$

So where did they get that matrix from? Any explanation is very much appreciated.

PS: I'm sorry this was meant to go to the Linear Algebra section!