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.
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!
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.
$$ \begin{bmatrix} 1 & * & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix} $$
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.
dCode has a page on Gauss elimination method, its result is a reduction of the starting matrix.
dCode retains ownership of the "Matrix Reduced Row Echelon Form" source code. Except explicit open source licence (indicated Creative Commons / free), the "Matrix Reduced Row Echelon Form" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, 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, or API access for "Matrix Reduced Row Echelon Form" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.
The copy-paste of the page "Matrix Reduced Row Echelon Form" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Matrix Reduced Row Echelon Form on dCode.fr [online website], retrieved on 2024-12-03,