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Math Help - Solving an abstract system of equations.

  1. #1
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    Solving an abstract system of equations.

    \text{Given }x=cy+bz,y=az+cx,z=bx+ay\text{ where } x,y,z
    \text{ are not all zero, prove that }a^2+b^2+c^2+2abc=1.

    Our professor hinted that the brute force way of solving this is naive and that there is a simpler way, the problem is I don't see it. My idea was to set this into a matrix with 3 equations and 3 unknowns and try to solve x, y, and z. This approach started to become very messy and I would get an answer like y=\frac{1-b^2}{a+cb}z. So my question is really this, is there anybody out there that sees the trick for this problem that could give me a pointer. thanks in advance!
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  2. #2
    A Plied Mathematician
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    Think of the vector \langle x,y,z\rangle^{T}. Then arrange your system thus:

    \begin{bmatrix}1&-c&-b\\<br />
c&-1&a\\<br />
b&a&-1\end{bmatrix}<br />
\begin{bmatrix}x\\y\\z\end{bmatrix}=<br />
\begin{bmatrix}0\\0\\0\end{bmatrix}.

    Naturally, the trivial solution is a solution of this system. However, by assumption, you've ruled that possibility out (x,y,z are not all zero). Therefore, there must be at least two solutions to the system. What does that tell you about the matrix

    \begin{bmatrix}1&-c&-b\\<br />
c&-1&a\\<br />
b&a&-1\end{bmatrix}?
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  3. #3
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    ok so we know
    that is it not full rank or that one of the rows is dependent on another row
    or that we will have infinitely many solutions to the homogenous system
    which means we have a free variable
    am i on the right track here in using this information to finish problem?
    would i solve the system with the bottom row zeroed out then see whats going on?
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  4. #4
    A Plied Mathematician
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    Well, you could solve the system. But you could also try using a condition that you know is true of any square system with an infinite number of solutions. Is the matrix singular or invertible?
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  5. #5
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    I SEE! its a det trick =) <3
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  6. #6
    A Plied Mathematician
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    That's right.
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