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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 a number of zeros starting from the left on each line increasing line by line, up to a complete line of zeros.

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# 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 $\begin{bmatrix} 1 & 2 & 4 \end{bmatrix} - \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 \end{bmatrix}$.

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