A rectangular matrix is in echelon form if

  1. All nonzero rows are above any rows of all zeros.
  2. Each leading entry of a row is in a column to the right of the leading entry of the row above it.
  3. All entries in a column below a leading entry are zero.

The matrix is in reduced echelon form if it satisfies the above requirements and also

  1. The leading entry in each row is one.
  2. The entries above each leading entry are zero as well.

Examples (x represents any nonzero entry, while * represents any entry):

x * * * *
0 0 0 x *
0 0 0 0 0

is in echelon form, while

1 * * 0 *
0 0 0 1 *
0 0 0 0 0

is in reduced echelon form.