# Gauss-Jordan elimination over any field

While it’s typical to solve a system of linear equations in real numbers, it’s also possible to solve a linear system over any mathematical field. Here is Java and Python code that defines various fields and provides a version of Gauss-Jordan elimination that works on any field. The basic Gauss-Jordan elimination algorithm can be adapted to solve systems of linear equations, invert matrices, calculate determinants, calculate ranks, and more.

## Source code ### Java version

Core classes:

• Field.java: Defines the constants and operations that every field must support.

• Matrix.java: Represents a matrix and implements a collection of operations:

• Basic information: Dimensions, get/set cell, clone, transpose

• Row operations: Swap rows, multiply row, add two rows, multiply two matrices

• Advanced operations: Gauss-Jordan elimination, determinant, inverse

Runnable classes:

• DemoMain.java: A short demonstration program, which implements the “Rational numbers” example below.

• MatrixTest.java: A JUnit test suite for Matrix.reducedRowEchelonForm(), Matrix.determinantAndRef(), and Matrix.invert() – mostly using PrimeField(11).

Included fields:

### Python version

• fieldmath.py: The library, including the classes Field, RationalField, PrimeField, BinaryField, QuadraticSurdField, QuadraticSurd, Matrix.

• demo-main.py: A short demonstration program, which implements the “Rational numbers” example below.

• matrix-test.py: A runnable test suite for fieldmath.Matrix.reduced_row_echelon_form(), fieldmath.Matrix.determinant_and_ref(), and fieldmath.Matrix.invert() – mostly using fieldmath.PrimeField(11).

## Examples

Rational numbers

Suppose we want to solve this system of linear equations in rational numbers:

$$\begin{cases} 2x + 5y + 3z &=& 7 \\ x + z &=& 1 \\ -4x + 2y - 9z &=& 6 \end{cases}$$

First we convert the system into an augmented matrix:

$$\begin{bmatrix} 2 & 5 & 3 & 7 \\ 1 & 0 & 1 & 1 \\ -4 & 2 & -9 & 6 \end{bmatrix}$$

Then we perform Gauss-Jordan elimination on the matrix:

$$\begin{bmatrix} 1 & 0 & 0 & 67/27 \\ 0 & 1 & 0 & 35/27 \\ 0 & 0 & 1 & -40/27 \end{bmatrix}$$

And as long as the left side is the identity matrix, we can read the answer off the rightmost column:

$$\begin{cases} x &=& 67/27 \\ y &=& 35/27 \\ z &=& -40/27 \end{cases}$$

Here is the code to perform this example:

// Nice-looking matrix
int[][] input = {
{2, 5, 3, 7},
{1, 0, 1, 1},
{-4, 2, -9, 6},
};

// The actual matrix object
Matrix<Fraction> mat = new Matrix<Fraction>(
input.length, input.length, RationalField.FIELD);
for (int i = 0; i < mat.rowCount(); i++) {
for (int j = 0; j < mat.columnCount(); j++)
mat.set(i, j, new Fraction(input[i][j], 1));
}

// Gauss-Jordan elimination
mat.reducedRowEchelonForm();
Integers modulo a prime

Suppose we want to solve this system of linear equations modulo 7:

$$\begin{cases} 3x + 1y + 4z &≡& 1 \mod 7 \\ 5x + 2y + 6z &≡& 5 \mod 7 \\ 0x + 5y + 2z &≡& 1 \mod 7 \end{cases}$$

First we convert the system into an augmented matrix:

$$\begin{bmatrix} 3 & 1 & 4 & 1 \\ 5 & 2 & 6 & 5 \\ 0 & 5 & 2 & 1 \end{bmatrix}$$

Then we perform Gauss-Jordan elimination on the matrix:

$$\begin{bmatrix} 1 & 0 & 0 & 4 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 0 \end{bmatrix}$$

And as long as the left side is the identity matrix, we can read the answer off the rightmost column:

$$\begin{cases} x &≡& 4 \mod 7 \\ y &≡& 3 \mod 7 \\ z &≡& 0 \mod 7 \end{cases}$$

Here is the code to perform this example:

// Nice-looking matrix
int[][] input = {
{3, 1, 4, 1},
{5, 2, 6, 5},
{0, 5, 2, 1},
};

// The actual matrix object
Matrix<Integer> mat = new Matrix<Integer>(
input.length, input.length, new PrimeField(7));
for (int i = 0; i < mat.rowCount(); i++) {
for (int j = 0; j < mat.columnCount(); j++)
mat.set(i, j, input[i][j]);
}

// Gauss-Jordan elimination
mat.reducedRowEchelonForm();