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

296 Chapter 8 Matrix algebra
Consider the system
AX=0
If |A| = 0,the systemhas non-trivial solutions.
If |A| ≠ 0,the systemhas only the trivialsolution.
Example8.40 Decide which of the following systems of equations has non-trivial solutions:
(a) 3x+7y=0
2x−y=0
(b) 2x+y=0
6x+3y=0
Solution (a) We writethe systemas
( ) ( 3 7 x 0
=
2 −1)(
y 0)
Let
Then
A =
( ) 3 7
2 −1
|A| = 3(−1) −2(7) = −17
Since the determinant ofAisnon-zero, the systemhas only the trivialsolution.
(b) We writethe systemas
( ( ) ( 2 1 x 0
=
6 3)
y 0)
Let
A =
( ) 2 1
6 3
fromwhich |A| = 2(3) −1(6) = 0.SincethedeterminantofAis0,thesystemhas
non-trivial solutions.
Example8.41 Determine which of the following systems ofequations has non-trivial solutions:
(a) 2x+y−3z=0
x−3y+2z=0
5x−8y+3z=0
(b) 2x+y−3z=0
x−3y+2z=0
5x−7y+3z=0