LU decomposition of $$$\left[\begin{array}{cccc}1 & 0 & 0 & 0\\0 & \frac{1}{34} & 0 & 0\\0 & 0 & 1 & 2\\0 & 0 & 2 & 1\end{array}\right]$$$

Find the LU decomposition of $$$\left[\begin{array}{cccc}1 & 0 & 0 & 0\\0 & \frac{1}{34} & 0 & 0\\0 & 0 & 1 & 2\\0 & 0 & 2 & 1\end{array}\right]$$$.

Start from the identity matrix $$$L = \left[\begin{array}{cccc}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{array}\right]$$$.

Subtract row $$$3$$$ multiplied by $$$2$$$ from row $$$4$$$: $$$R_{4} = R_{4} - 2 R_{3}$$$.

$$$\left[\begin{array}{cccc}1 & 0 & 0 & 0\\0 & \frac{1}{34} & 0 & 0\\0 & 0 & 1 & 2\\0 & 0 & 0 & -3\end{array}\right]$$$

Write the coefficient $$$2$$$ in the matrix $$$L$$$ at row $$$4$$$, column $$$3$$$:

$$$L = \left[\begin{array}{cccc}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 2 & 1\end{array}\right]$$$

The obtained matrix is the matrix $$$U$$$.

$$$L = \left[\begin{array}{cccc}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 2 & 1\end{array}\right]$$$A

$$$U = \left[\begin{array}{cccc}1 & 0 & 0 & 0\\0 & \frac{1}{34} & 0 & 0\\0 & 0 & 1 & 2\\0 & 0 & 0 & -3\end{array}\right]\approx \left[\begin{array}{cccc}1 & 0 & 0 & 0\\0 & 0.029411764705882 & 0 & 0\\0 & 0 & 1 & 2\\0 & 0 & 0 & -3\end{array}\right]$$$A