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

## 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|

## 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 cancelled out the -2 in the second square brackets, was I wrong?

Thanks for any hints and help :)

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SammyS
Staff Emeritus
Homework Helper
Gold Member
2-2λ+1) can be factored.

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

(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
Staff Emeritus
Homework Helper
Gold Member
(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.