Answer
The LU factorization:
$\mathbf{A}=\mathbf{L}\mathbf{U}=\begin{bmatrix}1&0&0\\-2&1&0\\-3&-5&1\end{bmatrix}\begin{bmatrix}-5&3&4\\0&-2&-1\\0&0&9\end{bmatrix}$
Work Step by Step
To find an LU factorization of the matrix
$\mathbf{A}=\begin{bmatrix}-5&3&4\\10&-8&-9\\15&1&2\end{bmatrix}$
Lets Reduce the rows of the matrix;
Let $R_{2}=R_{2}+2R_{1}$
$\mathbf{A}=\begin{bmatrix}-5&3&4\\0&-2&-1\\15&1&2\end{bmatrix}$
Let $R_{3}=R_{3}+3R_{1}$
$\mathbf{A}=\begin{bmatrix}-5&3&4\\0&-2&-1\\0&10&14\end{bmatrix}$
Let $R_{3}=R_{3}+5R_{2}$
$\mathbf{A}=\begin{bmatrix}-5&3&4\\0&-2&-1\\0&0&9\end{bmatrix}$
Hence,
$\mathbf{U}=\begin{bmatrix}-5&3&4\\0&-2&-1\\0&0&9\end{bmatrix}$
To find the matrix L we diivide each column by its pivot entry;
$\mathbf{L}=\begin{bmatrix}\frac{-5}{-5}&0&0\\\frac{10}{-5}&\frac{-2}{-2}&0\\\frac{15}{-5}&\frac{10}{-2}&\frac{9}{9}\end{bmatrix}=\begin{bmatrix}1&0&0\\-2&1&0\\-3&-5&1\end{bmatrix}$
$\mathbf{A}=\mathbf{L}\mathbf{U}=\begin{bmatrix}1&0&0\\-2&1&0\\-3&-5&1\end{bmatrix}\begin{bmatrix}-5&3&4\\0&-2&-1\\0&0&9\end{bmatrix}$
