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.
Matrix Reduced Row Echelon Form - dCode
Tag(s) : Matrix
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!
Matrix Reduced Row Echelon Form
Echelon Form Matrix Reduction Calculator
Answers to Questions (FAQ)
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" source code. Except explicit open source licence (indicated CC / Creative Commons / free), the "Matrix Reduced Row Echelon Form" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or the "Matrix Reduced Row Echelon Form" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, copy-paste, or API access for "Matrix Reduced Row Echelon Form" are not public, same for offline use on PC, tablet, iPhone or Android ! Remainder : dCode is free to use.
Need Help ?
Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!
Questions / Comments
Thanks to your feedback and relevant comments, dCode has developed the best 'Matrix Reduced Row Echelon Form' tool, so feel free to write! Thank you!