Row Echelon Form / Reduced Row Echelon Form

Table of contents
  1. Row Echelon Form
  2. Reduced Row Echelon Form

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

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).