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 nonzero 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
nonzero 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 



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 nonzero 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 nonzero 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 



A_{rref} 

B_{rref} 

C_{rref} 
Test Your Understanding
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 nonzero entry. This is a nono.
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 nonzero 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
nonzero element.
 The leading entry in each row is the only nonzero entry in its column.