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Fact 1 (scaling operation) If we replace an equation in a linear system by a nonzero multiple
of the equation, the solution set remains unchanged.
Ex.2 Let


2x1 −4x2 +2x3 =
0
0
2x2 −8x3 =
8
I.

−4x1 +5x2 +9x3 = −9
and


x1 −2x2 +x3 =
0
x2 −4x3 =
4 ,
I .

−4x1 +5x2 +9x3 = −9
00
then both I0 and I00 are equivalent to I. In other words, systems I, I0 , and I00 are essentially the same.
Notation:
Fact 2 (interchange operation) If we interchange two equations in a system, the solution set
remains unchanged.
Ex.3 Consider


x1 −2x2 +x3 =
0
I . −4x1 +5x2 +9x3 = −9 ,

2x2 −8x3 =
8
000
then I000 is equivalent to I.
Notation:
Fact 3 (replacement operation) If we replace one equation in a system by the sum of itself
and a multiple of another equation, the solution set remains unchanged.
Ex.4 The system

+x3 =
0
 x1 −2x2
2x2 −8x3 =
8
I(iv) .

−3x2 +13x3 = −9
is obtained by replacing
Notation:
of system I by
of I.
Definition Three operations described above are called elementary row operations. Thus elementary row operations do NOT alter the solution set of the original system.