math111_logo Row Echelon Form

3. Computing row echelon form

The leading entries in a row echelon form are called the pivots. The rows and the columns containing the pivots are the pivot rows and the pivot columns. For example, in the row echelon form

[ 1 3 4 5 ]
0 0 2 6
0 0 0 0

the pivots are the entries 1, 2, the pivot rows are [row 1] and [row 2], the pivot columns are [col 1] and [col 3]. Moreover, since we may use row operations to change the matrix

A = [ -1 -3 0 7 ]
1 3 2 -1
1 3 4 5

to the row echelon form above, we also say the pivot rows of A are [row 1] and [row 2], and the pivot columns of A are [col 1] and [col 3].

Example The following is the augmented matrix of a system studied before.

[ 3 1 -1 2 ]
1 -1 1 2
2 2 1 6

To find its row echelon form, we simplify one column at a time. We may create two zeros in [col 1] by (-3)[row 2] + [row 1] and (-2)[row 2] + [row 3].

[ 0 4 -4 -4 ]
1 -1 1 2
0 4 -1 2

Then the exchange [row 1] ↔ [row 2] moves the longest row to the top.

[ 1 -1 1 2 ]
0 4 -4 -4
0 4 -1 2

Now [col 1] cannot be further improved, and we turn to [col 2]. By (-1)[row 2] + [row 3], we have

[ 1 -1 1 2 ]
0 4 -4 -4
0 0 3 6

This is a row echelon form.

Example Consider a more complicated example.

[ 1 3 2 0 1 0 ]
-1 -1 -1 1 0 1
0 4 2 4 3 3
1 3 2 -2 0 0

To simplify [col 1], we first apply [row 2] + [row 4] and then apply [row 1] + [row 2] to get

[ 1 3 2 0 1 0 ]
0 2 1 1 1 1
0 4 2 4 3 3
0 2 1 -1 0 1

Now [col 1] is simplest, and we try to simplify [col 2]. More precisely, we simplify the more heavily shaded 3 by 5 submatrix. By (-1)[row 4] + [row 2] and (-2)[row 4] + [row 3], we get

[ 1 3 2 0 1 0 ]
0 0 0 2 1 0
0 0 0 6 3 1
0 2 1 -1 0 1

By exchanging [row 2], [row 3], and [row 4], we get

[ 1 3 2 0 1 0 ]
0 2 1 -1 0 1
0 0 0 2 1 0
0 0 0 6 3 1

Note that [col 2] is now simplest. It turns out that [col 3] is also simplest. Therefore we should try to simplify [col 4], or the more heavily shaded 2 by 3 submatrix. By (-3)[row 3] + [row 4], we get

[ 1 3 2 0 1 0 ]
0 2 1 -1 0 1
0 0 0 2 1 0
0 0 0 0 0 1

This is a row echelon form.

We remark that different choices of row operations may produce different matrices at the end. However, the shape of the row echelon form is independent of the choices (for a rigorous proof, see here).


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