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

8.11 Eigenvalues and eigenvectors 297
Solution (a) We have
AX=0
where
⎛ ⎞ ⎛ ⎞
2 1−3 x
A = ⎝1−3 2⎠ X = ⎝y⎠
5−8 3 z
Evaluation of |A|shows that |A| = 0 and so the systemhas non-trivial solutions.
(b) Here
⎛ ⎞
2 1−3
A = ⎝1−3 2⎠
5−7 3
from which |A| = −7.Since |A| ≠ 0 the systemhas only the trivialsolution.
EXERCISES8.11.1
1 Explain what ismeant by the trivial solution ofa
systemoflinear equations and what is meantby a
non-trivial solution.
2 Determine which ofthe following systemshas
non-trivial solutions:
(a) x−2y=0
3x−6y=0
(b) 3x+y=0
9x+2y=0
(c) 4x−3y=0
−4x+3y=0
(d) 6x−2y=0
2x − 2 3 y = 0
(e) y=2x
x=3y
3 Determine whichofthe following systemshave
non-trivial solutions:
(a) x+2y− z=0
3x+ y+2z=0
x+y =0
(b) 2x−3y−2z=0
3x+ y−3z=0
x−7y− z=0
(c) x+2y+3z=0
4x−3y− z=0
6x+ y+3z=0
(d) x+ 3z=0
x−y =0
y+2z=0
Solutions
2 (a),(c)and(d)have non-trivialsolutions. 3 (a)and(b)have non-trivialsolutions.
8.11.2 Eigenvalues
We will explain the meaning of the term eigenvalue by means of an example. Consider
the system
2x+y =λx
3x+4y=λy