# Calculator for NxN equation systems

## Gaussian elemination solver NxN

The calculator uses the Gaussian elemination to transform the matrix step-by-step in row echelon form. If the extended coefficient matrix brought to row echelon form by means of elementary transformations, the solution can be read directly.

$\left(\begin{array}{c}1\phantom{\rule{1em}{0ex}}{a}_{12}^{*}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{1n}^{*}\\ 0\phantom{\rule{2em}{0ex}}1\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{2n}^{*}\\ ⋮\\ 0\phantom{\rule{1em}{0ex}}0\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}0\phantom{\rule{1em}{0ex}}1\end{array}|\begin{array}{c}{b}_{1}^{*}\\ {b}_{2}^{*}\\ ⋮\\ {b}_{n}^{*}\end{array}\right)$

Linear Equation System NxN

$\begin{array}{c}{a}_{11}{x}_{1}+{a}_{12}{x}_{2}+\dots +{a}_{1n}{x}_{n}={b}_{1}\\ {a}_{21}{x}_{1}+{a}_{22}{x}_{2}+\dots +{a}_{2n}{x}_{n}={b}_{2}\\ ⋮\\ {a}_{m1}{x}_{1}+{a}_{m2}{x}_{2}+\dots +{a}_{mn}{x}_{n}={b}_{n}\end{array}$

or in matrix notation

$\left(\begin{array}{c}{a}_{11}\phantom{\rule{1em}{0ex}}{a}_{12}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{1n}\\ {a}_{21}\phantom{\rule{1em}{0ex}}{a}_{22}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{2n}\\ ⋮\\ {a}_{m1}\phantom{\rule{1em}{0ex}}{a}_{m2}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{mn}\end{array}\right)\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ ⋮\\ {x}_{n}\end{array}\right)=\left(\begin{array}{c}{b}_{1}\\ {b}_{2}\\ ⋮\\ {b}_{n}\end{array}\right)$

The schematic matrix form is used to show the transformations:

$\left(A|b\right)=\left(\begin{array}{c}{a}_{11}\phantom{\rule{1em}{0ex}}{a}_{12}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{1n}\\ {a}_{21}\phantom{\rule{1em}{0ex}}{a}_{22}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{2n}\\ ⋮\\ {a}_{m1}\phantom{\rule{1em}{0ex}}{a}_{m2}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{mn}\end{array}|\begin{array}{c}{b}_{1}\\ {b}_{2}\\ ⋮\\ {b}_{n}\end{array}\right)$

Dimension of the equation system N =
Number of digits =

Enter the coefficients: a11, a12, ... und b1, ...

## Solution with the Gauss algorithm

Step by step performing of the Gaussian elimination to get the row echelon form of the matrix.

The entered matrix is:

### The solution vector

The solution of the linear equation system is now given in the right column.

## More Calculators

Here is a list of of further useful calculators and sites:

Index Linear Equations Linear Equation Systems Calculator 2x2 systems Calculator 3x3 systems Calculator NxN Cramer's rule Calculator NxN Gauss method Matrix Determinant