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Math Help - Help with question on matrices

  1. #1
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    Help with question on matrices

    Hello
    My daughter is doing a college assignment and I managed to help her with the majority of the question but cannot figure out the last part.

    I have the characteristic equation and have made that a cubic but unsure on the M^-1 part
    Thanks
    Last edited by Jane1947; April 11th 2012 at 05:25 AM.
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  2. #2
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    Re: Help with question on matrices

    ok, this is what i have, but you may want to check my arithmetic, because i DO make mistakes:

    the Cayley-Hamilton theorem tells us M satisfies its own characteristic equation, which is (x+1)(x-2)(x-4) = x^3 - 5x^2 + 2x + 8.

    therefore: M^3 - 5M^2 + 2M + 8I = 0, or put another way:

    M^3 = 5M^2 - 2M - 8I.

    the trouble is now how to express M^2 in terms of M^{-1},M,I. to accomplish that, we write:

    M^3 - 5M^2 + 2M = -8I

    \frac{-1}{8}(M^2 - 5M + 2I)(M) = I, so that evidently:

    M^{-1} = \frac{-1}{8}(M^2 - 5M + 2I), or:

    -8M^{-1} = M^2 - 5M + 2I, and re-arranging:

    M^2 = 5M - 2I - 8M^{-1}.

    substituting back in our original equation:

    M^3 = 5M^2 - 2M - 8I = 5(5M - 2I - 8M^{-1}) - 2M - 8I

     = 23M - 18I -40M^{-1}, obtaining:

    p = 23, q = -18, r = -40.
    Thanks from a tutor and Jane1947
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  3. #3
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    Re: Help with question on matrices

    Thanks very much for the time and effort you have put into this. I appreciate the answer and shall sit down with a pencil and try and explain it to her!
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