LU decomposition of $$$\left[\begin{array}{ccccc}1 & 2 & 3 & 1 & 2\\1 & 4 & 2 & 3 & 1\\2 & 2 & -1 & 1 & 1\end{array}\right]$$$
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Find the LU decomposition of $$$\left[\begin{array}{ccccc}1 & 2 & 3 & 1 & 2\\1 & 4 & 2 & 3 & 1\\2 & 2 & -1 & 1 & 1\end{array}\right]$$$.
Solution
Start from the identity matrix $$$L = \left[\begin{array}{ccc}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{array}\right]$$$.
Subtract row $$$1$$$ from row $$$2$$$: $$$R_{2} = R_{2} - R_{1}$$$.
$$$\left[\begin{array}{ccccc}1 & 2 & 3 & 1 & 2\\0 & 2 & -1 & 2 & -1\\2 & 2 & -1 & 1 & 1\end{array}\right]$$$
Write the coefficient $$$1$$$ in the matrix $$$L$$$ at row $$$2$$$, column $$$1$$$:
$$$L = \left[\begin{array}{ccc}1 & 0 & 0\\1 & 1 & 0\\0 & 0 & 1\end{array}\right]$$$
Subtract row $$$1$$$ multiplied by $$$2$$$ from row $$$3$$$: $$$R_{3} = R_{3} - 2 R_{1}$$$.
$$$\left[\begin{array}{ccccc}1 & 2 & 3 & 1 & 2\\0 & 2 & -1 & 2 & -1\\0 & -2 & -7 & -1 & -3\end{array}\right]$$$
Write the coefficient $$$2$$$ in the matrix $$$L$$$ at row $$$3$$$, column $$$1$$$:
$$$L = \left[\begin{array}{ccc}1 & 0 & 0\\1 & 1 & 0\\2 & 0 & 1\end{array}\right]$$$
Add row $$$2$$$ to row $$$3$$$: $$$R_{3} = R_{3} + R_{2}$$$.
$$$\left[\begin{array}{ccccc}1 & 2 & 3 & 1 & 2\\0 & 2 & -1 & 2 & -1\\0 & 0 & -8 & 1 & -4\end{array}\right]$$$
Write the coefficient $$$-1$$$ in the matrix $$$L$$$ at row $$$3$$$, column $$$2$$$:
$$$L = \left[\begin{array}{ccc}1 & 0 & 0\\1 & 1 & 0\\2 & -1 & 1\end{array}\right]$$$
The obtained matrix is the matrix $$$U$$$.
Answer
$$$L = \left[\begin{array}{ccc}1 & 0 & 0\\1 & 1 & 0\\2 & -1 & 1\end{array}\right]$$$A
$$$U = \left[\begin{array}{ccccc}1 & 2 & 3 & 1 & 2\\0 & 2 & -1 & 2 & -1\\0 & 0 & -8 & 1 & -4\end{array}\right]$$$A