LU decomposition of $$$\left[\begin{array}{ccc}1 & 2 & 1\\3 & 4 & 2\\2 & 1 & 1\end{array}\right]$$$

Find the LU decomposition of $$$\left[\begin{array}{ccc}1 & 2 & 1\\3 & 4 & 2\\2 & 1 & 1\end{array}\right]$$$.

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

Subtract row $$$1$$$ multiplied by $$$3$$$ from row $$$2$$$: $$$R_{2} = R_{2} - 3 R_{1}$$$.

$$$\left[\begin{array}{ccc}1 & 2 & 1\\0 & -2 & -1\\2 & 1 & 1\end{array}\right]$$$

Write the coefficient $$$3$$$ in the matrix $$$L$$$ at row $$$2$$$, column $$$1$$$:

$$$L = \left[\begin{array}{ccc}1 & 0 & 0\\3 & 1 & 0\\0 & 0 & 1\end{array}\right]$$$

Subtract row $$$1$$$ multiplied by $$$2$$$ from row $$$3$$$: $$$R_{3} = R_{3} - 2 R_{1}$$$.

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

Write the coefficient $$$2$$$ in the matrix $$$L$$$ at row $$$3$$$, column $$$1$$$:

$$$L = \left[\begin{array}{ccc}1 & 0 & 0\\3 & 1 & 0\\2 & 0 & 1\end{array}\right]$$$

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

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

Write the coefficient $$$\frac{3}{2}$$$ in the matrix $$$L$$$ at row $$$3$$$, column $$$2$$$:

$$$L = \left[\begin{array}{ccc}1 & 0 & 0\\3 & 1 & 0\\2 & \frac{3}{2} & 1\end{array}\right]$$$

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

$$$L = \left[\begin{array}{ccc}1 & 0 & 0\\3 & 1 & 0\\2 & \frac{3}{2} & 1\end{array}\right] = \left[\begin{array}{ccc}1 & 0 & 0\\3 & 1 & 0\\2 & 1.5 & 1\end{array}\right]$$$A

$$$U = \left[\begin{array}{ccc}1 & 2 & 1\\0 & -2 & -1\\0 & 0 & \frac{1}{2}\end{array}\right] = \left[\begin{array}{ccc}1 & 2 & 1\\0 & -2 & -1\\0 & 0 & 0.5\end{array}\right]$$$A