Echelon Form of a Matrix

This lesson introduces the concept of an echelon matrix. Echelon matrices come 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 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.

Each of the matrices shown below are examples of matrices in row echelon form.

1 2 3 4
0 0 1 3
0 0 0 1
 
1 2 3 4
0 0 1 3
0 0 0 1
0 0 0 0
 
1 2
0 1
0 0
Aref   Bref   Cref

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.

Reduced Row Echelon Form

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

Each of the matrices shown below are examples of matrices in reduced row echelon form.

1 2 0 0
0 0 1 0
0 0 0 1
 
1 2 0 0
0 0 1 0
0 0 0 1
0 0 0 0
 
1 0
0 1
0 0
Arref   Brref   Crref

Test Your Understanding of This Lesson

Problem 1

Which of the following matrices is in row echelon form?

0 1
1 0
0 0
1 2
0 1
0 0
1 2
0 1
0 1
1 0
0 0
0 1
A B C D

(A) Matrix A
(B) Matrix B
(C) Matrix C
(D) Matrix D
(E) None of the above

Solution

The correct answer is (B), since it satisfies all of the requirements for a row echelon matrix. The other matrices fall short.

  • The leading entry in Row 1 of matrix A is to the right of the leading entry in Row 2, which is inconsistent with definition of a row echelon matrix.

  • In matrix C, the leading entries in Rows 2 and 3 are in the same column, which is not allowed.

  • In matrix D, the row with all zeros (Row 2) comes before a row with a non-zero entry. This is a no-no.

Problem 2

Which of the following matrices are in reduced row echelon form?

1 0 0 0
0 0 1 0
0 0 0 1
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
0 1 0 0
0 0 0 1
0 0 0 0
0 0 0 0
A B C

(A) Only matrix A
(B) Only matrix B
(C) Only matrix C
(D) All of the above
(E) None of the above

Solution

The correct answer is (D), since each matrix satisfies all of the requirements for a reduced row echelon matrix.

  • 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.
  • The leading entry in each row is the only non-zero entry in its column.