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Math Help - Determinant of a Vandermonde matrix

  1. #1
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    Determinant of a Vandermonde matrix

    Consider the 3x3 Vandermonde matrix:

    V=[1 x1 x1^2; 1 x2 x2^2; 1 x3 x3^2]

    Show that the det(V) = (x2-x1)(x3-x1)(x3-x2)

    Sorry for the format. I did R2-R1 and R3-R1 then found the det to be (x2-x1)(x3^2-x1)-(x2^2-x1^2)(x3-x1)

    Is this just an algebraic nightmare or did I do something wrong? Thank you!
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  2. #2
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    Quote Originally Posted by pakman View Post
    Consider the 3x3 Vandermonde matrix:

    V=[1 x1 x1^2; 1 x2 x2^2; 1 x3 x3^2]

    Show that the det(V) = (x2-x1)(x3-x1)(x3-x2)

    Sorry for the format. I did R2-R1 and R3-R1 then found the det to be (x2-x1)(x3^2-x1)-(x2^2-x1^2)(x3-x1)

    Is this just an algebraic nightmare or did I do something wrong? Thank you!
    You've done something wrong. The answer is -(x_1 - x_3) (x_1 - x_2) (x_2 - x_3).

    I hope you took the easy road and expanded along the first column.
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    You've done something wrong. The answer is -(x_1 - x_3) (x_1 - x_2) (x_2 - x_3).

    I hope you took the easy road and expanded along the first column.
    That's what I did. I subtracted row 1 from both row 2 and 3 to get 0's in a21 and a31. But then it gets a bit messy from there
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  4. #4
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    Quote Originally Posted by pakman View Post
    Consider the 3x3 Vandermonde matrix:

    V=[1 x1 x1^2; 1 x2 x2^2; 1 x3 x3^2]

    Show that the det(V) = (x2-x1)(x3-x1)(x3-x2)

    Sorry for the format. I did R2-R1 and R3-R1 then found the det to be (x2-x1)(x3^2-x1)-(x2^2-x1^2)(x3-x1)

    Is this just an algebraic nightmare or did I do something wrong? Thank you!
    You should have (x2 - x1)(x3^2 - x1^2) - (x2^2 - x1^2)(x3 - x1) ....

    = (x2 - x1)(x3 - x1)(x3 + x1) - (x2 - x1)(x2 + x1)(x3 - x1)

    = (x2 - x1)(x3 - x1) [(x3 + x1) - (x2 + x1)] = ......
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  5. #5
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    Ahh okay, stupid algebraic mistake. Thanks!
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