382 CHAPTER 18. LINEAR FUNCTIONS
Now denote by Pi j the elementary matrix which comes from the identity from switchingrows i and j. From what was just explained and Proposition 18.1.24,
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 18.2.5 Let Pi j denote the elementary matrix which involves switching the ith
and the jth rows. ThenPi jA = B
where B is obtained from A by switching the ith and the jth rows.
Example 18.2.6 Consider the following. 0 1 01 0 00 0 1
a bg de f
=
g da be f
Next consider the row operation which involves multiplying the ith row by a nonzero
constant, c. The elementary matrix which results from applying this operation to the ith rowof the identity matrix is of the form
. . . 01
c1
0. . .
Now consider what this does to a column vector.
. . . 01
c1
0. . .
...vi−1vi
vi+1...
=
...vi−1cvi
vi+1...
Denote by E (c, i) this elementary matrix which multiplies the ith row of the identity by thenonzero constant, c. Then from what was just discussed and Proposition 18.1.24,
E (c, i)
......
...a(i−1)1 a(i−1)2 · · · a(i−1)p
ai1 ai2 · · · aipa(i+1)1 a(i+1)2 · · · a(i+1)p
......
...
=
......
...a(i−1)1 a(i−1)2 · · · a(i−1)p
cai1 cai2 · · · caipa(i+1)1 a(i+1)2 · · · a(i+1)p
......
...