Search for a tool
Matrix Reduced Row Echelon Form

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

Results

Matrix Reduced Row Echelon Form -

Tag(s) : Matrix

Share dCode and you

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!

Please, check our community Discord for help requests!

Thanks to your feedback and relevant comments, dCode has developped the best Matrix Reduced Row Echelon Form tool, so feel free to write! Thank you !

# Matrix Reduced Row Echelon Form

## Echelon Form Matrix Reduction Calculator

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

### What is a matrix in row echelon form?

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 matrix $$\begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 8 \\ 0 & 0 & 0 \end{bmatrix}$$ can be reduced in a matrix echelon form $$\begin{bmatrix} 1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}$$ in two steps : 1/ Multiplication of row 2 by 1/2 (or division by 2) $\begin{bmatrix} 2 & 4 & 8 \end{bmatrix}$ becomes $\begin{bmatrix} 1 & 2 & 4 \end{bmatrix}$ and 2/ subtraction of row 2 to row 1 $\begin{bmatrix} 1 & 2 & 4 \end{bmatrix} - \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 \end{bmatrix}$.

## Source code

dCode retains ownership of the online 'Matrix Reduced Row Echelon Form' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) no data, script or API access will be for free, same for Matrix Reduced Row Echelon Form download for offline use on PC, tablet, iPhone or Android !

## Need Help ?

Please, check our community Discord for help requests!