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Math Help - Matrix Algebra 3

  1. #1
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    Matrix Algebra 3

    The question is:

    Given that dim(m) = 2x2 and M^2 - M + 2I =0, show that M^4 + 3M^2 + 4I =0

    It doesn't say what M is, though
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  2. #2
    MHF Contributor red_dog's Avatar
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    M^2-M+2I=0\Rightarrow M^3=M^2-2M

    (M^2-M+2I)^2=0

    M^4-2M^3+5M^2-4M+4I=0

    Replace M^3:

    M^4-2M^2+4M+5M^2-4M+4I=0\Rightarrow M^4+3M^2+4I=0
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  3. #3
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    Why does M^3 = M^2 - 2M ?
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  4. #4
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    You are told that M^2- M+ 2I= 0. Adding M- 2I to both sides, M^2= M- 2I. Now multiply both sides by M to get M^3= M^2- 2M
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  5. #5
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    Thank you! But I don't understand the rest of the working either?

    Why is (M^2 - M + 2I)^2 =0?

    And could you explain the rest of it too ,please?
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  6. #6
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    Quote Originally Posted by juliak View Post
    Thank you! But I don't understand the rest of the working either?

    Why is (M^2 - M + 2I)^2 =0?

    And could you explain the rest of it too ,please?
    If A = 0*I then A^2 = A*A = 0*I*0*I = 0*I

    The rest is simply technical.
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