Answer
$L=\left[\begin{array}{cccc}1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 4 & 5 & 1 & 0 \\ -2 & -1 & 0 & 1\end{array}\right], U=\left[\begin{array}{cccc}1 & 3 & -5 & -3 \\ 0 & -2 & 3 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{array}\right]$
Work Step by Step
Dicrease matrix $A$ to row echelon form which represents matrix $U$.
$\left[\begin{array}{cccc}\underline{1} & 3 & -5 & -3 \\ \frac{-1}{4} & -5 & 8 & 4 \\ \frac{4}{2} & 2 & -5 & -7 \\ -2 & -4 & 7 & 5\end{array}\right]$, multiply first row with 1,-4,2 and add to second, third
and fourth
$\sim\left[\begin{array}{cccc}1 & 3 & -5 & -3 \\ 0 & \frac{-2}{0} & 3 & 1 \\ 0 & \frac{-10}{15} & 15 & 5 \\ 0 & \underline{2} & -3 & -1\end{array}\right],$ multiply second row with -5,1 and add to third,
$\sim\left[\begin{array}{cccc}1 & 3 & -5 & -3 \\ 0 & -2 & 3 & 1 \\ 0 & 0 & \underline{0} & 0 \\ 0 & 0 & 0 & \underline{0}\end{array}\right]=U$
Put underlined elements (use 1 instead zero) into matrix.
$\left[\begin{array}{cccc}1 & 0 & 0 & 0 \\ -1 & -2 & 0 & 0 \\ 4 & -10 & 1 & 0 \\ -2 & 2 & 0 & 1\end{array}\right]$
Now to get matrix $L$ divide each column with leading element. First column with $1,$ second with -2 and third with 1 .
$L=\left[\begin{array}{cccc}1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 4 & 5 & 1 & 0 \\ -2 & -1 & 0 & 1\end{array}\right]$
