Verify that i is an eigenvalue of A and that xi is a corresp

Verify that _i is an eigenvalue of A and that x_i is a corresponding eigenvector.

₁ = 4, x₁ = (1, 0, 0)

₂ = 2, x₂ = (1, 2, 0)

₃ = 3, x₃ = (1, 1, 1)

Ax₁

₁x₁

Ax₂

₂x₂

Ax₃

₃x₃

A = 4 1 2 0 2 1 0 0 3 ,

1 = 4, x1 = (1, 0, 0) 2 = 2, x2 = (1, 2, 0) 3 = 3, x3 = (1, 1, 1)

Solution

The matrix A is

4

-1

2

0

2

1

0

0

3

If is an eigenvalue of A, then we have det(A- I₃) = 0. Further, det(A- I₃) = - ³+9²-26+24.

Further Ax₁ = (4,0,0)^T = 4(1,0,0)^T = 4x₁, hence x₁= (1,0,0)^T is an eigenvector of A corresponding to its eigenvalue ₁= 4.

Also, Ax₂ = (2,4,0)^T = 2(1,2,0)^T = 2x₂, hence x₂= (1,2,0)^T is an eigenvector of A corresponding to its eigenvalue ₂= 2.

Further, Ax₃ = (-3,3,3)^T = 3(-1,1,1)^T = 3x₃, hence x₃= (-1,1,1)^T is an eigenvector of A corresponding to its eigenvalue ₃= 3.