18.2. ROW OPERATIONS AND LINEAR EQUATIONS 383

This proves the following lemma.

Lemma 18.2.7 Let E (c, i) denote the elementary matrix corresponding to the row op-eration 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.

Example 18.2.8 Consider this. 1 0 00 5 00 0 1

 a bc de f

=

 a b5c 5de f

Finally consider the third of these row operations. Denote by E (c× i+ j) the elemen-

tary matrix which replaces the jth row with the jth row added to c times the ith row. In casei < j this will be of the form 

. . . 01

. . .c 1

0. . .

Now consider what this does to a column vector.

. . . 01

. . .c 1

0. . .





...vi...

v j...

=



...vi...

cvi + v j...

Now from this and Proposition 18.1.24,

E (c× i+ j)



......

...ai1 ai2 · · · aip...

......

a j1 a j2 · · · a jp...

......



=



......

...ai1 ai2 · · · aip...

......

cai1 +a j1 cai2 +a j2 · · · caip +a jp...

......

The case where i > j is handled similarly. This proves the following lemma.