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Math Help - Determinant

  1. #1
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    Determinant

    Let a_1,\cdots , a_n be given numbers. Compute the determinant of the nxn matrix A=(a_{ij}), where a_{ij}=a^{i-1}_j.


    I don't understand what is meant by this: a_{ij}=a^{i-1}_j
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  2. #2
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    Re: Determinant

    without being given any more context, i would assume it means a_{ij} is the (i-1)-th power of a_j. so the first row should all be 1's. this kind of matrix is called a Vandermonde matrix.

    you'll probably arrive at a better understanding if you try it first for n = 2, and n = 3. try to factor the result into binomials.

    if you are clever, and figure out (guess) the general form, you can actually prove it by induction on n.
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  3. #3
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    Re: Determinant

    Quote Originally Posted by Deveno View Post
    without being given any more context, i would assume it means a_{ij} is the (i-1)-th power of a_j. so the first row should all be 1's. this kind of matrix is called a Vandermonde matrix.

    you'll probably arrive at a better understanding if you try it first for n = 2, and n = 3. try to factor the result into binomials.
    That is the question 100% verbatim.
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  4. #4
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    Re: Determinant

    I see it as a^{i-1}_j \in \{a_1,\cdots , a_n\}
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    Re: Determinant

    well it's an important matrix, both for the study of polynomials, alternating bilinear forms, and permutation groups. it crops up in a lot of different places. time to roll up your sleeves and make some messy calcs...lol

    @pickslides: i don't think so....there's nothing in the wording to indicate membership. and the calculation of such a determinant (a Vandermonde matrix) is the sort of thing you might encounter in a variety of different courses, inclusing differential geometry, linear algebra and group theory. i think the first time i saw it was in a physics class.
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    Re: Determinant

    Quote Originally Posted by Deveno View Post
    well it's an important matrix, both for the study of polynomials, alternating bilinear forms, and permutation groups. it crops up in a lot of different places. time to roll up your sleeves and make some messy calcs...lol

    I have used it before in my undergraduate linear alg. class. I am familiar with it. I just don't know it when I am presented with it.
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  7. #7
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    Re: Determinant

    V\ =\ \begin{bmatrix}1&1&\cdots&1\\a_1&a_2&\cdots&a_n\\a  _1^2&a_2^2&\cdots&a_n^2\\ \vdots&\vdots&\ddots&\vdots\\a_1^{n-1}&a_2^{n-1}&\cdots&a_n^{n-1} \end{bmatrix}

    does it look better written out like this?
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  8. #8
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    Re: Determinant

    Quote Originally Posted by Deveno View Post
    V\ =\ \begin{bmatrix}1&1&\cdots&1\\a_1&a_2&\cdots&a_n\\a  _1^2&a_2^2&\cdots&a_n^2\\ \vdots&\vdots&\ddots&\vdots\\a_1^{n-1}&a_2^{n-1}&\cdots&a_n^{n-1} \end{bmatrix}

    does it look better written out like this?
    I understood when you set it was a Vandermonde matrix. That wasn't needed.
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  9. #9
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    Re: Determinant

    fair enough. but perhaps if someone else peruses this thread one day, it will be helpful to them. no slight intended.
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