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Math Help - 3x3 matricies proof

  1. #1
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    3x3 matricies proof

    Assuming that det(AB) = det(A)det(B) for 3x3 matricies, prove that...

    | a+b+c , a^2+b^2+c^2 , a^3+b^3+c^3|
    |a^2+b^2+c^2 , a^3+b^3+c^3 , a^4+b^4+c^4| = abc(b-c)^2(c-a)^2(a-b)^2
    |a^3+b^3+c^3 , a^4+b^4+c^4 , a^5+b^5+c^5|

    The problem I encounter with this problem is that I either find that I get an answer of 0 or I end up with a very large equation which I can't find a way to cancel. Any help would be apreciated.

    Rodregez
    Last edited by Therodregez; November 18th 2009 at 06:42 PM.
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  2. #2
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    Multiply A=\left(\begin{array}{ccc}1&1&1\\a&b&c\\a^{2}&b^{2  }&c^{2}\\\end{array}\right) and B=\left(\begin{array}{ccc}a&a^{2}&a^{3}\\b&b^{2}&b  ^{3}\\c&c^{2}&c^{3}\\\end{array}\right) and then use the given identity.

    The determinants of A and B factor easily using elementary operations.
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  3. #3
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    Quote Originally Posted by BobP View Post
    Multiply A=\left(\begin{array}{ccc}1&1&1\\a&b&c\\a^{2}&b^{2  }&c^{2}\\\end{array}\right) and B=\left(\begin{array}{ccc}a&a^{2}&a^{3}\\b&b^{2}&b  ^{3}\\c&c^{2}&c^{3}\\\end{array}\right) and then use the given identity.

    The determinants of A and B factor easily using elementary operations.
    Thank you, im still struggling with this question though, I cant get determinant B to come out right, i figure det(A) = (c-b)(b-a)(a-c) which looks promising but i get det(B) = (c-b)(bc-a(c+b)+a^2) so im not sure how to proceed from there really.

    Thank you, Rodregez
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  4. #4
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    Factors

    For figuring out det(B), factor out an 'a' from the 1st line, a 'b' from the 2nd line and a 'c' from the 3rd. Then det(B) = abc det(new matrix). Note the new matrix is just A transposed.
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