• Support PF! Buy your school textbooks, materials and every day products Here!

[Linear Algebra] For which a is 0 an eigenvalue?

  • Thread starter Ortix
  • Start date
  • #1
64
0

Homework Statement



I have to find for which "a" an eigenvalue for the following system is 0.

The system:

1 -1 1
-1 2 -2
0 a 1

Homework Equations


My characterstic equation:
(1-λ)(2-λ)(1-λ)+2a -(1-λ) -a = 0


The Attempt at a Solution



I then proceed:
(1-λ)(λ2-3λ-2+a) = 0

but then i'm kind of clueless.. Now what?
 

Answers and Replies

  • #2
938
9
You wrote the equation for the eigenvalues of the system. Now, if you want 0 to be an eigenvalue, then it better satisfy that equation.
 
  • #3
64
0
Well I get a=2 but the answer is a=-1

Can't seem to find my error. I've tried it a bajillion times (3 times actually) :)
 
  • #4
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,916
19
(1-λ)(λ2-3λ-2+a) = 0
Where did this come from?
 
  • #5
938
9
Can you check for the 4th time, what happens if you plug in λ=0 to (1-λ)(2-λ)(1-λ)+2a -(1-λ) -a = 0 ? :) You just calculated something wrong somewhere along the way. (I'm assuming the characteristic equation is correct)
 
  • #6
HallsofIvy
Science Advisor
Homework Helper
41,833
955
You really don't need to find the entire eigenvalue equation to answer this. A matrix has 0 as an eigenvalue if and only if it is NOT invertible (since there must be a non-zero v such that Av= 0) and that is true if and only if its determinant is 0. Set the determinant, which depends on a, equal to 0 and solve for a.
 
  • #7
64
0
HallsofIvy, you tha man! Solved it! :D
 
  • #8
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,728

Homework Statement



I have to find for which "a" an eigenvalue for the following system is 0.

The system:

1 -1 1
-1 2 -2
0 a 1

Homework Equations


My characterstic equation:
(1-λ)(2-λ)(1-λ)+2a -(1-λ) -a = 0


The Attempt at a Solution



I then proceed:
(1-λ)(λ2-3λ-2+a) = 0

but then i'm kind of clueless.. Now what?
The equation in (a) is not consistent with that in (b). You don't need (b); just plug λ=0 into (a).

RGV
 

Related Threads on [Linear Algebra] For which a is 0 an eigenvalue?

  • Last Post
Replies
5
Views
1K
Replies
4
Views
1K
Replies
1
Views
861
  • Last Post
Replies
1
Views
752
Replies
5
Views
1K
Replies
1
Views
3K
  • Last Post
Replies
8
Views
1K
  • Last Post
Replies
3
Views
921
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
2
Views
905
Top