114 CHAPTER 4. ROW OPERATIONS

Denote by E (c, i) this elementary matrix which multiplies the ith row of the identityby the nonzero constant, c. Then from what was just discussed and the way matrices aremultiplied,

E (c, i)



a11 a12 · · · · · · a1p...

......

ai1 ai2 · · · · · · aip...

......

an1 an2 · · · · · · anp

equals a matrix having the columns indicated below.

=

E (c, i)



a11...

ai1...

an1

, E (c, i)



a12...

ai2...

an2

, · · · , E (c, i)



a1p...

aip...

anp





=



a11 a12 · · · · · · a1p...

......

cai1 cai2 · · · · · · caip...

......

an1 an2 · · · · · · anp

This proves the following lemma.

Lemma 4.1.4 Let E (c, i) denote the elementary matrix corresponding to the row opera-tion in which the ith row is multiplied by the nonzero constant, c. Thus E (c, i) involvesmultiplying the ith row of the identity matrix by c. Then

E (c, i)A = B

where B is obtained from A by multiplying the ith row of A by c.

Finally consider the third of these row operations. Denote by E (c× i+ j) the elementarymatrix which replaces the jth row with itself added to c times the ith row added to it. Incase i < j this will be of the form

1 0. . .

1...

. . .

c · · · 1. . .

0 1

