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.
A matrix is in reduced row echelon form (rref)
when it satisfies the following conditions.
The matrix is in row echelon form (i.e., it satisfies the
three conditions listed above).
The leading entry in each row is the only non-zero entry in
its column.
A matrix in echelon form is called an echelon matrix.
Matrix A and matrix B are examples
of echelon matrices.
1
2
3
4
0
0
1
3
0
0
0
1
0
0
0
0
1
2
0
0
0
0
1
0
0
0
0
1
0
0
0
0
A
B
Matrix A is in row echelon form, and matrix B
is in reduced row echelon form.
How to Transform a Matrix Into Its Echelon Forms
Any matrix can be transformed into its echelon forms, using a series of
elementary row operations. Here's how.
Pivot the matrix
Find the pivot, the first non-zero entry in
the first column of the matrix.
Interchange rows, moving the pivot row to the first row.
Multiply each element in the pivot row by the inverse of
the pivot, so the pivot equals 1.
Add multiples of the pivot row to each of the lower rows,
so every element in the pivot column of the lower
rows equals 0.
To get the matrix in row echelon form, repeat the pivot
Repeat the procedure from Step 1 above, ignoring previous
pivot rows.
Continue until there are no more pivots to be
processed.
To get the matrix in reduced row echelon form, process non-zero
entries above each pivot.
Identify the last row having a pivot equal to 1, and let
this be the pivot row.
Add multiples of the pivot row to each of the upper rows,
until every element above the pivot equals 0.
Moving up the matrix, repeat this process for each row.
Transforming a Matrix Into Its Echelon Forms: An Example
To illustrate the transformation process,
let's transform Matrix A to a row echelon
form and to a reduced row echelon form.
0
1
2
1
2
1
2
7
8
⇒
1
2
1
0
1
2
2
7
8
⇒
1
2
1
0
1
2
0
3
6
A
A_{1}
A_{2}
⇒
1
2
1
0
1
2
0
0
0
⇒
1
0
-3
0
1
2
0
0
0
A_{ref}
A_{rref}
To transform matrix A into its echelon forms, we implemented
the following series of elementary row operations.
We found the first non-zero entry in the first column
of the matrix in row 2; so we interchanged Rows 1 and 2, resulting
in matrix A_{1}.
Working with matrix A_{1},
we multiplied each element of Row 1 by -2 and added the result to
Row 3. This produced A_{2}.
Working with matrix A_{2},
we multiplied each element of Row 2 by -3 and added the result to
Row 3. This produced A_{ref}. Notice that
A_{ref} is in row echelon form, because it
meets the following requirements: (a) the first non-zero entry of each
row is 1, (b) the first non-zero entry is to the right of the first
non-zero entry in the previous row, and (c) rows made up entirely of
zeros are at the bottom of the matrix.
And finally, working with matrix A_{ref},
we multiplied the second row by -2 and added it to
the first row. This produced A_{rref}.
Notice that A_{rref} is in reduced
row echelon form, because it satisfies the requirements for
row echelon form plus each leading non-zero entry is the only non-zero
entry in its column.
Note: The row echelon matrix that results from a series of
elementary row operations is not necessarily unique. A different set of row
operations could result in a different row echelon matrix. However,
the reduced row echelon matrix is unique; each matrix has only one
reduced row echelon matrix.
Test Your Understanding
Problem 1
Consider the matrix X, shown below.
X =
0
1
1
2
0
5
Which of the following matrices is the reduced row echelon form of
matrix X ?
0
1
1
0
0
0
1
0
0
1
0
0
1
0
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). The elementary row operations used to change
Matrix X into its reduced row echelon form are shown
below.
0
1
1
2
0
5
⇒
1
2
0
1
0
5
⇒
1
2
0
1
0
0
⇒
1
0
0
1
0
0
X
X_{1}
X_{2}
X_{rref}
To change X to its reduced row echelon form, we take the
following steps:
Interchange Rows 1 and 2, producing X_{1}.
In X_{1},
multiply Row 2 by -5 and add it to Row 3, producing
X_{2}.
In X_{2},
multiply Row 2 by -2 and add it to Row 1, producing
X_{rref}.
Note: Matrix A is not in reduced row echelon form, because
the leading entry in Row 2 is to the left of the leading entry in
Row 3; it should be to the right. Matrix C
is not in reduced row echelon form, because column 2 has more
than one non-zero entry. And finally, matrix D is not
in reduced row echelon form, because Row 2 with all zeros is followed
by a row with a non-zero element; all-zero rows must follow non-zero rows.