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Math Help - need example of a 3x3 non-zero matrix B and a polynomial p(x)

  1. #1
    Newbie Nona's Avatar
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    need example of a 3x3 non-zero matrix B and a polynomial p(x)

    Hello,
    I need help with:
    Give an example of a 3x3 non-zero matrix B and a polynomial p(x) of degree 2 such that P(B) = 0.
    Thank you
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  2. #2
    MHF Contributor

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    Every polynomial satisfies it own "minimal equation" so I would suggest that you start with the equation and work backwards.

    For example, start with x^2- 3x+ 2= (x- 1)(x- 2)= 0. Any matrix satisfying A^2- 3A+ 2I= 0 must have 1 and 2 as eigenvalues. Since you want a 3 by 3 matrix, take either one of those as a "double" eigenvalue and construct a matrix having eigenvalues 1, 1, 2, or 1, 2, 2. And, since you are only asked for a single example, a diagonal matrix is easiest to construct.
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