Find the eigenvalues and their corresponding eigenvectors. Please help!!!

Question

Find the eigenvalues and their corresponding eigenvectors of:

  

Hint: Use EROs to create two zeros in one row or one column of      first.

Answer 1

det(B - λE) = (1 - λ)²(5 - λ) + 8 + 8 - 4(5 - λ) + 4(1 - λ) + 4(1 - λ) = 0;

(λ² - 2λ - 3)(5 - λ) + 16 + 8(1 - λ) = 0; 5λ² - 10λ - 15 - λ³ + 2λ² + 3λ + 24 - 8λ = 0;λ

λ³ - 7λ² + 15λ - 9 = 0

Using rational zeros therem: lisr of possible zeros ± 1, ±3, ±9.

And we have λ = 1, λ = 3 and λ = 3.

So igenvectors: for λ = 1 [(0 2 2) (-2 4 2) (2 - 2 0)][x y z] = [0 0 0] and we get system:-2y + 2z = 0, 2x + 4y - 2z = 0, 2x + 2y = 0 so y = - x, z = -x and igenvector for λ = 1 is (-1, 1, 1).

If λ = 3 we have [(-2 2 2) (- 2 2 2) (2 - 2 - 2)]{x y z] = [0 0 0] and we get 3 equations: -2x - 2y + 2z = 0,

2x + 2y - 2z = 0 and 2x + 2y - 2z = 0 or just one equation x + y - z = 0 and we have 2 igenvectors:

(1, 0, 1) and (0, 1, 1)