082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

288 Chapter 8 Matrix algebra
The aim, as before, istocarryout rowoperations on the stepped form
( )
231
1 1 3
inordertoobtainvaluesofzerounderthestep.Clearlytoachievetherequiredform,the
row operations we performed earlier arerequired, thatis
( )
R 1 2 31
R 2
→2R 2
−R 1 0 −1 5
The last line means 0x −1y = 5, that isy = −5, and finally back substitution yieldsx,
as before.
This technique has other advantages in that it allows us to observe other forms of
behaviour.Weshallseethatsomeequationshaveauniquesolution,somehavenosolutions,whileothers
have an infinite number.
Example8.34 UseGaussian eliminationtosolve
2x+3y=4
4x+6y=7
Solution Inaugmentedmatrix form wehave
( )
234
4 6 7
We proceed toeliminate entries under the step:
( )
R 1 23 4
R 2
→R 2
−2R 1 0 0 −1
Study of the last line seems to imply that 0x + 0y = −1, which is clearly nonsense.
When this happens the equations have no solutions and we say that the simultaneous
equations areinconsistent.
Example8.35 UseGaussian eliminationtosolve
x+y=0
2x+2y=0
Solution Inaugmentedmatrix form wehave
( )
110
2 2 0
Eliminating entries under the stepwefind
( )
R 1 110
R 2
→R 2
−2R 1 0 0 0
This last line implies that 0x + 0y = 0. This is not an inconsistency, but we are now
observing a third type of behaviour. Whenever this happens we need to introduce what
are called free variables. The first row starts off with a non-zero x. There is now no
row which starts off with a non-zero y. We therefore say y is free and choose it to be