# LU Decomposition Calculator

The calculator will find (if possible) the LU decomposition of the given matrix $A$, i.e. such a lower triangular matrix $L$ and an upper triangular matrix $U$ that $A=LU$, with steps shown.

In case of partial pivoting (permutation of rows is needed), the calculator will also find the permutation matrix $P$ such that $PA=LU$.

Related calculator: QR Factorization Calculator

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Find the LU decomposition of $\left[\begin{array}{ccc}2 & 7 & 1\\3 & -2 & 0\\1 & 5 & 3\end{array}\right]$.

## 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 $1$ multiplied by $\frac{3}{2}$ from row $2$: $R_{2} = R_{2} - \frac{3 R_{1}}{2}$.

$\left[\begin{array}{ccc}2 & 7 & 1\\0 & - \frac{25}{2} & - \frac{3}{2}\\1 & 5 & 3\end{array}\right]$

Write the coefficient $\frac{3}{2}$ in the matrix $L$ at row $2$, column $1$:

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

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

$\left[\begin{array}{ccc}2 & 7 & 1\\0 & - \frac{25}{2} & - \frac{3}{2}\\0 & \frac{3}{2} & \frac{5}{2}\end{array}\right]$

Write the coefficient $\frac{1}{2}$ in the matrix $L$ at row $3$, column $1$:

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

Add row $2$ multiplied by $\frac{3}{25}$ to row $3$: $R_{3} = R_{3} + \frac{3 R_{2}}{25}$.

$\left[\begin{array}{ccc}2 & 7 & 1\\0 & - \frac{25}{2} & - \frac{3}{2}\\0 & 0 & \frac{58}{25}\end{array}\right]$

Write the coefficient $- \frac{3}{25}$ in the matrix $L$ at row $3$, column $2$:

$L = \left[\begin{array}{ccc}1 & 0 & 0\\\frac{3}{2} & 1 & 0\\\frac{1}{2} & - \frac{3}{25} & 1\end{array}\right]$

The obtained matrix is the matrix $U$.

$L = \left[\begin{array}{ccc}1 & 0 & 0\\\frac{3}{2} & 1 & 0\\\frac{1}{2} & - \frac{3}{25} & 1\end{array}\right] = \left[\begin{array}{ccc}1 & 0 & 0\\1.5 & 1 & 0\\0.5 & -0.12 & 1\end{array}\right]$A
$U = \left[\begin{array}{ccc}2 & 7 & 1\\0 & - \frac{25}{2} & - \frac{3}{2}\\0 & 0 & \frac{58}{25}\end{array}\right] = \left[\begin{array}{ccc}2 & 7 & 1\\0 & -12.5 & -1.5\\0 & 0 & 2.32\end{array}\right]$A