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Math Help - Imagenary cube root Problems

  1. #1
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    Exclamation Imagenary cube root Problems

    If w is an imagenary cube root of unity , then find the value of

    <br />
\begin{vmatrix}<br />
1+w & w^2  & -w \\<br />
1+w^2 & w & -w^2 \\<br />
w^2+w & w & -w^2<br />
\end{vmatrix}<br />
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  2. #2
    Super Member Gamma's Avatar
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    did you type it right? that bottom 2x2 minor has rows the same, is that correct?

    If so, just do it, i recommend cofactor down the first column to take advantage of that being the same.

    the first term is 0.

    second is
    -(1+w^2)[-w+w^2]
    third is
    (w^2+w)[-w+w^2]

    Add them
    (-1-w^2 +w^2 +w)[-w+w^2]=(-1+w)[-w+w^2]=1-w^2-w^2+w=-2w^2+w+1
    But these are cube roots of unity, w^2+w=-1 \Rightarrow w=-1-w^2


    -2w^2+w+1= -2w^2 (-1 - w^2) + 1= -3w^3=-3
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  3. #3
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    I didnt get the last part

    Quote Originally Posted by Gamma View Post
    did you type it right? that bottom 2x2 minor has rows the same, is that correct?

    If so, just do it, i recommend cofactor down the first column to take advantage of that being the same.

    the first term is 0.

    second is
    -(1+w^2)[-w+w^2]
    third is
    (w^2+w)[-w+w^2]

    Add them
    (-1-w^2 +w^2 +w)[-w+w^2]=(-1+w)[-w+w^2]=1-w^2-w^2+w=-2w^2+w+1
    But these are cube roots of unity, w^2+w=-1 \Rightarrow w=-1-w^2


    -2w^2+w+1= -2w^2 (-1 - w^2) + 1= -3w^3=-3
    I didnt get the last part
    -2w^2+w+1= -2w^2 (-1 - w^2) + 1= -3w^3=-3

    could u please explain how u got this?????
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  4. #4
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    i have some doubt

    Quote Originally Posted by zorro View Post
    I didnt get the last part
    -2w^2+w+1= -2w^2 (-1 - w^2) + 1= -3w^3=-3

    could u please explain how u got this?????
    -Firstly the -3w^3 shouldnt it be -3w^2

    -Secondly where did the w^2 go and how did -3w^2 \ = \ -3
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  5. #5
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    Could u please provide me with links

    Quote Originally Posted by zorro View Post
    -Firstly the -3w^3 shouldnt it be -3w^2

    -Secondly where did the w^2 go and how did -3w^2 \ = \ -3

    Could u please provide me with links where i could get some examples of this type
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