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

[Ortix]

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?

 

[clamtrox]

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.

 

[Ortix]

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) :)

 

[Hurkyl]

(1-λ)(λ²-3λ-2+a) = 0

Where did this come from?

 

[clamtrox]

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)

 

[HallsofIvy]

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.

 

[Ortix]

HallsofIvy, you tha man! Solved it! :D

 

[Ray Vickson]

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