Statistics Dictionary

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Reduced Row Echelon Form

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.

Note: Some references present a slightly different description of the row echelon form. They do not require that the first non-zero entry in each row is equal to 1.

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