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
1 2 3 4 1 2
0 0 1 3
0 0 1 3 0 1
0 0 0 1
0 0 0 1 0 0
0 0 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 ent
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
1 2 0 0 1 0
0 0 1 0
0 0 1 0 0 1
0 0 0 1
0 0 0 1 0 0
0 0 0 0
Arref Brref Crref
Test Your Understanding
Problem 1
Which of the following matrices is in row echelon form?
0 1 1 2 1 2 1 0
1 0 0 1 0 1 0 0
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
1 2 3 4 1 2
0 0 1 3
0 0 1 3 0 1
0 0 0 1
0 0 0 1 0 0
0 0 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 ent
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
1 2 0 0 1 0
0 0 1 0
0 0 1 0 0 1
0 0 0 1
0 0 0 1 0 0
0 0 0 0
Arref Brref Crref
Test Your Understanding
Problem 1
Which of the following matrices is in row echelon form?
0 1 1 2 1 2 1 0
1 0 0 1 0 1 0 0
