Consider the matrix A = [[2, 0, 0],[1,1,-2],[1,-1, 0]] (c) For the values of k in (a), ii. Verify that A(vector v_k)= k(vector v_k). These are called Eigenvectors of A. Here are the links to the...

Consider the matrix

A = [[2, 0, 0],[1,1,-2],[1,-1, 0]]

(c) For the values of k in (a),

ii. Verify that A(vector v_k)= k(vector v_k). These are called Eigenvectors of A.

Here are the links to the other parts of the problem :

http://www.enotes.com/homework-help/consider-matrix-2-0-0-1-1-2-1-1-0-find-430769

http://www.enotes.com/homework-help/consider-matrix-2-0-0-1-1-2-1-1-0-b-which-values-430771

Expert Answers
jeew-m eNotes educator| Certified Educator

VAlues for K obtained from prevoius parts is;

K = -1 and K = 2.

When k=-1:

`[[2-(-1), 0, 0],[1, 1-(-1),-2],[1,-1,-(-1)]]`

= [[3, 0, 0],[1, 2,-2],[1,-1,1]]

`= [[1, 0, 0],[1, 2,-2],[1,-1,1]]`

`= [[1, 0, 0],[0, 2,-2],[0,-1,1]]`

Vector `v_(-1) = [0,1,1]^T ` is an eigenvector of -1

`Av_(-1) = [[2, 0, 0],[1,1,-2],[1,-1, 0]] [0,1,1]^T `

`= [0, -1, -1]T`

`= -v_(-1)`

When k=2:

`[[2-(2), 0, 0],[1, 1-(2),-2],[1,-1,-(2)]]`

`= [[0, 0, 0],[1, -1,-2],[1,-1,-2]]`

`= [[0, 0, 0],[1, -1,-2],[0,0,0]]`

Vectors `v2 =[1, 1, 0]^T` and `v(2') = [2,0,1]^T` are both eigenvectors

`Av_2= [[2, 0, 0],[1,1,-2],[1,-1, 0]] [1,1,0]^T `

`=[2, 2, 0]T`

`= 2v_2`

`Av_(2') = [[2, 0, 0],[1,1,-2],[1,-1, 0]] [2,0,1]^T` 

`= [4, 0, 2]^T`

`= 2v_(2')`

From the previous parts we can get k = -1 and k = 2.

When k=-1;

Vector v-1 = [0,1,1]^T  is an eigenvector of -1

When k=2

Vectors  and are both eigenvectors

Hence it is proved that A(vector v_k)= k(vector v_k)