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I came across a problem of eigenvalues and eigenvectors. It was easy and I solved it but one thing made me unsure about the answer. All the three eigenvectors were zero vectors. Here is the question and my answer:

The matrix A=

( -1 0 0 1

0 -2 0 0

0 1 -2 0

0 0 0 1)

I began with finding the eigenvalues. The result is the following fourth order equation (x=lamda=eigenvalue):

x^4+4x^3+x^2-6x-4=0

When i solved this equation using Texas Instruments calculator, it found three solutions:

x1=-(sqrt(5)+1)

x2=(sqrt(5)-1)

x3= -1

x4= -1

So, I had three eignevectors because x3=x4=-1 (multiplicity 2)

When I used these eigenvalues to find the eigenvectors, all of the eigenvectors turned out to be=

(0

0

0

0)

My question: is this reasonable solution to have all of the eigenvectors= 0 or I have made a mistake somewhere.

Thanks in advance

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# Eigenvalues and eigenvectors

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