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Math Help - matrix

  1. #1
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    matrix

    Show that  \det \begin{bmatrix} 1 & 1 & 1 \\ a & b & c \\ a^{2} & b^{2} & c^{2} \end{bmatrix} = (b-a)(c-a)(c-b) .

    Using expansion by minors I got  c^{2}(b-a) + b^{2}(a-c) + a^{2}(c-b) . How would I convert this to the above form?
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  2. #2
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    I think I got it.

    Thanks
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  3. #3
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    Quote Originally Posted by heathrowjohnny View Post
    Show that  \det \begin{bmatrix} 1 & 1 & 1 \\ a & b & c \\ a^{2} & b^{2} & c^{2} \end{bmatrix} = (b-a)(c-a)(c-b) .

    Using expansion by minors I got  c^{2}(b-a) + b^{2}(a-c) + a^{2}(c-b) . How would I convert this to the above form?
    = c^{2}(b-a) + b^2 a - b^2 c + a^2 c- a^2 b = c^{2}(b-a) + b^2 a - a^2 b - b^2 c + a^2 c

     = c^2 (b - a) + ab (b - a) - c (b^2 - a^2)

     = c^2 (b - a) + ab (b - a) - c (b - a)(b + a) =  (b - a)(c^2 + ab - cb - ca)

    = (b - a)(c^2 - cb + ab - ca) = (b - a)(c[c - b] - a[c - b]) = (b - a)(c - a)(c - b) .

    Edit: Well, for what it's worth .....
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