Questions related to Eigenvalues with 2 repeated roots and Eigenvectors, please show me the steps on how to answer the repeated roots in the easiest way possible thank you.

Question

Matrix of A (3x3) of the system AX = λX

A = [6 -2 2]

[-2 3 -1]

[2 -1 3]

a) Find the eigenvalues of A

b) Find the corresponding eigenvectors of A

Answer 1

So A-λI gives:

[6-λ -2 2]

[-2 3-λ -1]

[2 -1 3-λ]

Taking the determinant and setting to zero gives:

-λ³+12λ²-36λ+32 = 0

factors of 32 = 32,16,8,4,2,1

See if 2 is a root, ie f(2) = 0

-8 + 48 -72 +32 = 0 Yes

synthetic division

-1 12 -36 32 |2

-2 20 -32

---------------------

-1 10 -16 0

(λ-2)(-λ²+10λ-16) = (λ-2)(λ-2)(λ-8) = 0

λ = 2,2,8