Row Echelon Form / Reduced Row Echelon Form

Table of contents

1. Row Echelon Form
2. Reduced Row Echelon Form

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Row Echelon Form

A row echelon form matrix satisfies the following conditions:

A pivot entry is the first non-zero entry of a row in a row echelon form matrix.

1. All-zero rows are at the bottom of the matrix.
2. All pivot entries are $1$.
3. The pivot entries of rows are strictly to the right of the pivot entries of rows above them.
4. All entries below a pivot entry are zero.

\[\begin{bmatrix} \mathbf{1} & 0 & 3 & 4 \\ 0 & \mathbf{1} & 2 & 3 \\ 0 & 0 & 0 & \mathbf{1} \\ 0 & 0 & 0 & 0 \\ \end{bmatrix}\]

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Reduced Row Echelon Form

A reduced row echelon form matrix satisfies the following conditions:

• All the conditions of a row echelon form matrix.
• The pivot entries are the only non-zero entries in their columns.

\[\begin{bmatrix} \mathbf{1} & 0 & 3 & 0 \\ 0 & \mathbf{1} & 2 & 0 \\ 0 & 0 & 0 & \mathbf{1} \\ 0 & 0 & 0 & 0 \\ \end{bmatrix}\]

Common methods to find the reduced row echelon form of a matrix are Gaussian elimination and Gauss-Jordan elimination, which involve elementary row operations (row switching and linear combinations of rows).

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