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Math Help - determinant problem

  1. #1
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    determinant problem

    A=\left(\begin{array}{ccc}-3 & 1 & -1\\-7 & 5 & -1\\-6 & 6 & -2\end{array}\right)find the definative polinomial?
    if i go by definition
    |tI-A|=(t+3)[t^{2}-3t-4]+13t+20
    so i need to open the cols and find the roots of polinomial of power 3 which is alot of work
    the answer in the book is (t+2)^{2}(t-4)
    is there some other way to get this result
    ?
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  2. #2
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    Re: determinant problem

    check your arithmetic.

    for det(tI - A), i get (t+3)(t^2 - 3t - 4) + t - 4 = t^3 - 12t - 16.

    to factor t^3 - 12t - 16, you might try guessing rational (and thus integer) roots first, that is: t = -1,1,-2,2,-4,4,-8,8,-16,16.

    on the 3rd try, you would find that (-2)^3 - 12(-2) - 16 = -8 + 24 -16 = 0, so you would know that t+2 is a factor.

    division by t+2 will give you a quadratic, to which you could apply the quadratic formula (unless you could factor it by inspection).
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  3. #3
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    Re: determinant problem

    Noticing that "t+ 2" factor, I would have been inclined to expand the determinant on the third column!
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  4. #4
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    Re: determinant problem

    ok thanks
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