Eigenvalues and eigenvectors, 3x3 matrix using Remainder and Factor Theorem

[Jowin86]

Homework Statement

i = the 3x3 matrix below

2-λ 0 1
-1 4-λ -1
-1 2 0-λ

Using remainder and factor theorem find the 3 values of λ.

Homework Equations

|i| = a1|b2c3-c2b3|-a2|a2c3-c2a3|+a3|a2b3-b2a3|

|a|=ad-bc

The Attempt at a Solution

(2-λ) |(4-λ x 0-λ)-(-1x2)|+1|(-1x2)-(4-λ x -1) **because b1 is 0 I've left it out**

(2-λ)[(4-λ)(0-λ)+2] +1 [-2+(4-λ)]

(2-λ)(4-λ)(0-λ)+1(4-λ)

Factorise out (4-λ):

(4-λ)[(2-λ)(0-λ)+1]

Multiply [] brackets out (FOIL):

(4-λ)(λ²-2λ+1)

...and now I'm stuck which probably means I've gona wrong somewhere ;-(

*should I have taken out 0-λ?

*on the 2nd line of my attempted answer I figured the +2 in the first square brackets canceled out the -2 in the second square brackets, was I wrong?Thanks for any hints and help :)

 

[SammyS]

(λ²-2λ+1) can be factored.

If you can't do that, then find the roots: λ²-2λ+1 = 0

 

[Jowin86]

(4-λ) λ(-2λ)

λ=4 λ=2 λ=0

Is that wrong?

I'm not sure how to find the roots of λ²-2λ+1=0 ?

 

[SammyS]

(4-λ) λ(-2λ)

λ=4 λ=2 λ=0

Is that wrong?

Yes, it's wrong

I'm not sure how to find the roots of λ²-2λ+1=0 ?

It's a quadratic equation. Solve it.