Statistics Dictionary

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Echelon Matrix

Echelon matrices come in two forms: the row echelon form (ref) and the reduced row echelon form (rref).

A matrix is in row echelon form (ref) when it satisfies the following conditions.

  • The first non-zero element in each row, called the leading entry, is 1.
  • Each leading entry is in a column to the right of the leading entry in the previous row.
  • Rows with all zero elements, if any, are below rows having a non-zero element.

A matrix is in reduced row echelon form (rref) when it satisfies the following conditions.

  • The matrix is in row echelon form (i.e., it satisfies the three conditions listed above).
  • The leading entry in each row is the only non-zero entry in its column.

A matrix in echelon form is called an echelon matrix. Matrix A and matrix B are examples of echelon matrices.

1 2 3 4
0 0 1 3
0 0 0 1
0 0 0 0
 
1 2 0 0
0 0 1 0
0 0 0 1
0 0 0 0
A   B

Matrix A is in row echelon form, and matrix B is in reduced row echelon form.

See also:  Echelon Form of a Matrix | Changing a Matrix Into Echelon Form