Question

# Find the LU-Factorization of the matrix A below A=\begin{bmatrix}2 & 1 & -1 \\-2 & 0 & 3 & \\ 2 & 1 & -4\\4 & 1 & -4 \\ 6 & 5 & -2\end{bmatrix}

Polynomial factorization
Find the LU-Factorization of the matrix A below
$$A=\begin{bmatrix}2 & 1 & -1 \\-2 & 0 & 3 & \\ 2 & 1 & -4\\4 & 1 & -4 \\ 6 & 5 & -2\end{bmatrix}$$

2021-05-10
Step 1
Given
$$\begin{bmatrix}2 & 1 & -1 \\-2 & 0 & 3 & \\ 2 & 1 & -4\\4 & 1 & -4 \\ 6 & 5 & -2\end{bmatrix}$$
Step 2
Solution
L=Lower triangular matrix
U=Upper triangular matrix
$$A=\begin{bmatrix}2 & 1 & -1 \\-2 & 0 & 3 \\ 2 & 1 & -4\\4 & 1 & -4 \\ 6 & 5 & -2\end{bmatrix}\sim \begin{bmatrix}2 & 1 & -1 \\0 & 1 & 2 \\ 0 & 0 & 3\\0 & -1 & -2 \\ 0 & 2 & 1\end{bmatrix}\left(\begin{matrix}R_{2}\rightarrow & R_{2}+ & R_{1}\\R_{3}\rightarrow & R_{3}- & R_{1} \\ R_{4}\rightarrow & R_{4}- & 2R_{1}\\R_{5}\rightarrow & R_{5}- & 3R_{1}\end{matrix}\right)$$
$$\sim\begin{bmatrix}2 & 1 & -1 \\0 & 1 & 2 \\ 0 & 0 & -3\\0 & 0 & 0 \\ 0 & 0 & -3\end{bmatrix}R_{4}\rightarrow R_{4}+R_{2}, R_{5}\rightarrow R_{5}-2R_{2}$$
$$\sim\begin{bmatrix}2 & 1 & -1 \\0 & 1 & 2 \\ 0 & 0 & -3\\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}R_{5}\rightarrow R_{5}-R_{3}$$
$$U=\begin{bmatrix}2 & 1 & -1 \\0 & 1 & 2 \\ 0 & 0 & -3\\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$$
Step 3
And lower triangular matrix that is L is calculated by the entries are the number in the 0 respectively.
i.e., $$L=\begin{bmatrix}1 & 0 & 0 \\-1 & 1 & 0 \\ 1 & 0 & 1\\2 & -1 & 0 \\ 3 & 2 & 1\end{bmatrix}$$