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Matrix Reduced Row Echelon Form

Tool to reduce a matrix to its echelon row form (reduced). A row reduced matrix has an increasing number of zeros starting from the left on each row.

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# Matrix Reduced Row Echelon Form

## RREF with Modulo Calculator

### What is a matrix in row echelon form? (Definition)

An reduced row echelon form matrix (RREF) is a matrix of the form $$\begin{bmatrix} \oplus & * & * & * \\ 0 & 0 & \oplus & * \\ 0 & 0 & 0 & \oplus \\ 0 & 0 & 0 & 0 \end{bmatrix}$$

The $*$ are any coefficients and the $\oplus$ are non-zero coefficients called pivots.

A row reduced matrix is an echelon matrix whose pivots are 1 with coefficients in the column of the pivot equal to zero.

$$\begin{bmatrix} 1 & * & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$

### How to transform a matrix into an echelon matrix?

The transformation method of any matrix into a reduced row echelon matrix is possible by means of row operations such as:

— the permutation of 2 rows

— the multiplication of a row by a non-zero constant

— the addition of a row or a multiple of a row

Example: The following matrix $M$ can be reduced in row echelon form in 4 steps: $$M = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 2 & 4 & 8 \end{bmatrix}$$
1/ Permutation of row 2 and 3 to get a row 3 filled with $0$.
2/ Multiplication if the new row 2 by 1/2 (ou division par 2) : $\begin{bmatrix} 2 & 4 & 8 \end{bmatrix}$ becomes $\begin{bmatrix} 1 & 2 & 4 \end{bmatrix}$
3/ subtraction of row 1 to row 2: $\begin{bmatrix} 1 & 2 & 4 \end{bmatrix} - \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 \end{bmatrix}$
4/ subtraction of 3 times row 2 to row 1: $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix} - 3 \cdot \begin{bmatrix} 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 2 & 0 \end{bmatrix}$.
The matrix in row echelon form is $$\begin{bmatrix} 1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}$$

This method is called Gaussian elimination.

### How to echelon a matrix with Gauss method?

dCode has a page on Gauss elimination method, its result is a reduction of the starting matrix.

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