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    hi, basically i don't know where the [1, 0, 0] matrix came from, or where they got the following two matrices for multiplication. i understand how they went on from there but i'm just not getting the whole (A-x1I)(A-x2I) equation? any help? thank you!
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    For a general matrix A, suppose the eigenvectors are v1, v2, v3.

    Then we know (1, 0, 0) can be written as a1v1+a2v2+a3v3 for some scalars v1, v2, v3.

    Then multiplying by (A-x1I) "kills off" the a1v1 component (since A-x1I)v1 = 0.
    Extending this idea, multiplying by (A-x1I)(A-x2I) kills off the a1v1 and a2v2 components, leaving only a multiply of v3, the third eigenvector.

    [Note that there are various (uncommon) situations where this approach fails, either because v3 has the same eigenvalue as one of v1, v2 (in which case (A-x1I)(A-x2I) kills off v3 as well), or because the representation of (1, 0, 0) in terms of v1, v2, v3 has a3v3 = 0].
 
 
 
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