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

## 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?

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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.

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
Staff Emeritus
Gold Member
(1-λ)(λ2-3λ-2+a) = 0
Where did this come from?

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
Homework Helper
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.

HallsofIvy, you tha man! Solved it! :D

Ray Vickson
Homework Helper
Dearly Missed

## 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