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 |
| 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 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 |
| 0 |
0 |
0 |
0 |
|
|
|
| Arref |
|
Brref |
|
Crref |
Test Your Understanding of This Lesson
Problem 1
Which of the following matrices is in row echelon form?
(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.
|