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2.2: Elementary Row Operations 45
Example 2.2.3
Put the matrix
1 1 2 2
4 4 9 10
3 3 6 7
in reduced row echelon form.
By applying suitable row operations we find the row ech-
elon form to be
' 1 1 2 2'
0 0 1 2
0 0 0 1
Notice that columns 1, 3 and 4 contain pivots. To pass to
reduced row echelon form, apply the row operations Ri — 2R 2,
R\ + 2i?3 and R 2 — 2R3: the answer is
1 1 0 0'
0 0 1 0
0 0 0 1
As this example illustrates, one can pass from row echelon
form to reduced row echelon form by applying further row op-
erations; notice that this will not change the number of pivots.
Thus an arbitrary matrix can be put in reduced row echelon
form by applying a finite sequence of elementary row opera-
tions. The reader should observe that this is just the matrix
formulation of the Gauss-Jordan elimination procedure.
Exercises 2.2
1. Put each of the following matrices in row echelon form:
, (b)
