92 CHAPTER 4. MATRICES

The importance of elementary matrices is that when you multiply on the left by one, itdoes the row operation which was used to produce the elementary matrix.

Now consider what these elementary matrices look like. First consider the one whichinvolves switching row i and row j where i < j. This matrix is of the form

. . .

0 1. . .

1 0. . .

Note how the ith and jth rows are switched in the identity matrix and there are thus all oneson the main diagonal except for those two positions indicated. The two exceptional rowsare shown. The ith row was the jth and the jth row was the ith in the identity matrix. Nowconsider what this does to a column vector.

. . .

0 1. . .

1 0. . .





...xi...

x j...

=



...x j...xi...

Now denote by Pi j the elementary matrix which comes from the identity from switching

rows i and j. From what was just explained,

Pi j



......

...ai1 ai2 · · · aip...

......

a j1 a j2 · · · a jp...

......

=



......

...a j1 a j2 · · · a jp

......

...ai1 ai2 · · · aip...

......

This has established the following lemma.

Lemma 4.5.3 Let Pi j denote the elementary matrix which involves switching the ith andthe jth rows. Then

Pi jA = B

where B is obtained from A by switching the ith and the jth rows.

Example 4.5.4 Consider the following. 0 1 01 0 00 0 1

 a b

g de f

=

 g da be f

