Matrix Reduced Row Echelon Form

An echelon form matrix 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 of any matrix into a reduced row echelon matrix is possible by means of operations such as the permutation of 2 lines, the multiplication of a line by a non-zero constant or the addition of a line or a multiple of a line.

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 \( \begin{bmatrix} 1 & 2 & 4 \end{bmatrix} - \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 \end{bmatrix} \).