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Math Help - Determine the value of (1 + xyz)

  1. #1
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    Red face Determine the value of (1 + xyz)

    Question : If x, y, z are all different and given that
    <br />
\begin{vmatrix}<br />
x & x^2 & 1 + x^3 \\<br />
y & y^2 & 1 + y^3 \\<br />
z & z^2 & 1 + z^3<br />
\end{vmatrix} = 0,<br />

    Determine the value of (1 + xyz)
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  2. #2
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    What have you done, and where are you stuck?
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  3. #3
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    \det\begin{bmatrix} x&x^2&1+x^3\\ y&y^2&1+y^3\\ z&z^2&1+z^3 \end{bmatrix}=\cdots=(z-x)(z-y)(y-x)(1+xyz)=0, if you have enough patience.

    \because x\neq y\neq z\neq x \ \therefore 1+xyz=0
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  4. #4
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  5. #5
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    Could u please provide the end result

    Quote Originally Posted by mr fantastic View Post


    I have seen the reply post in the other thread but cold u please provide me with the end answer, just to be sure that i have done it right or no.At present i am trying to do as provided by the prev thread but it would be good if u could provide with the end result
    ie, (1 + xyz)
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  6. #6
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    Quote Originally Posted by zorro View Post
    I have seen the reply post in the other thread but cold u please provide me with the end answer, just to be sure that i have done it right or no.At present i am trying to do as provided by the prev thread but it would be good if u could provide with the end result
    ie, (1 + xyz)
    You need to start showing some effort here. Post all your work. Say where you get stuck.
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  7. #7
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    Is this correct?

    Quote Originally Posted by mr fantastic View Post
    You need to start showing some effort here. Post all your work. Say where you get stuck.

    After doing the determinant i am getting the following please advice if correct or no

    (xy^2 - xz^2 - yx^2 + zx^2 + yz^2 - zy^2)(1+xyz)=0

    \therefore (1+xyz) = 0
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  8. #8
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    Quote Originally Posted by zorro View Post
    After doing the determinant i am getting the following please advice if correct or no

    (xy^2 - xz^2 - yx^2 + zx^2 + yz^2 - zy^2)(1+xyz)=0

    \therefore (1+xyz) = 0
    Post #3 already confirms your answer.
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  9. #9
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    Quote Originally Posted by mr fantastic View Post
    Post #3 already confirms your answer.
    Just wanted to check i did it correctly or no .........want to know if the step is correct or no?????
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  10. #10
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    Quote Originally Posted by zorro View Post
    Just wanted to check i did it correctly or no .........want to know if the step is correct or no?????
    Look at post #3 and compare answers. Does (xy^2 - xz^2 - yx^2 + zx^2 + yz^2 - zy^2) (which is what you have) equal (z-x)(z-y)(y-x) ? (the answer in post #3 is correct, by the way).
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  11. #11
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    Quote Originally Posted by zorro View Post
    After doing the determinant i am getting the following please advice if correct or no

    (xy^2 - xz^2 - yx^2 + zx^2 + yz^2 - zy^2)(1+xyz)=0

    \therefore (1+xyz) = 0

    I think some marks will be deducted in your solution .

    You should mention that iff two of the elements are the same , the factor

     xy^2 - xz^2 - yx^2 + zx^2 + yz^2 - zy^2 = 0

    because you have only showed that

    xy^2 - xz^2 - yx^2 + zx^2 + yz^2 - zy^2 = 0

    OR

     1 + xyz = 0

    so  1 + xyz is not 100% zero

    Factorising it you will get full mark !
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