5 For which values of k is the vector 2 3 2 1 5 4 1 an eigen

5.) For which value(s) of k is the vector 2 3 2 1 5 -4 1 an eigenvector of the matrix A

Solution

Find eigenvalues from the characteristic polynomial:
|{{-3-, 1, -1}, {3, 2-, 1}, {5, -4, 1-}}|=-^3++6=(-^2-2*-3)(-2)
1=2, hence one of the eigen value is 2

now putting =2 in above matrices we find that

[{-5, 1,-1}, {3, 0,1},{5,-4,-1}] .[1 2 k ] = 0 since matrix dot eigen vector = 0

therefore

-5 +2 -k =0

hence k = -3

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