Please answer part (b) in great detail, mostly, how to calculate the 3rd vector

Question

Please answer part (b) in great detail, mostly, how to calculate the 3rd vector for the doubly degenerate value of lambda = 2 with a corresponding eigenvector of [1 0 -1]^t

Consider the matrix By performing a suitable matrix multiplication, show that X1 and X2 are eigenvectors of A, where X1 = [-2,1, 1]t and X2 = [-1,0, 1]1. What are the corresponding eigenvalues? Using your answer to (a), find the third eigenvalue and its corresponding eigenvector. Choose any constants so that X3 has simple entries and it is linearly independent to X1 and X2.

Explanation / Answer

eigen values for X1 and X2 are 1, 2 respectively.

let ? be an eigen value of A.Then by definition,

A X = ?X, where X is an eigen vector.

=> (A-?I)X = O.

=> |A-? I| = 0

=> -?^3+5 ?^2-8 ?+4 = 0.

=> (?-1)(?-2)^2 = 0

=> 1 and 2 are the only eigen values of A. so we cant find an eigen vector which is independent of X1 and X2.

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