How to determine whether matrices are in echelon form, reduced echelon form or not in echelon form?

A rectangular array of m x n numbers (real or complex) in the form of m horizontal lines (called rows) and n vertical lines (called columns), is called a matrix of order m by n, written as m x n matrix. Such an array is enclosed by [ ] or ().

Echelon form of matrix

Echelon matrices are in two forms: the row echelon form (REF) and the reduced row echelon form (RREF).

Row echelon form

A matrix is in row echelon form (REF) when it meets 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.

Reduced row echelon form

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

  • The matrix satisfies conditions for a row echelon form.
  • The leading entry in each row is the only non-zero entry in its column.

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