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
A_{ref}
B_{ref}
C_{ref}
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
A_{rref}
B_{rref}
C_{rref}
Test Your Understanding
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
A
B
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
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.