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Question

The constant $$k$$ is such that the following system of equations posses a non-trivial(i.e., not all zero) solution over the set of rationals $$Q$$  
$$ x+ky+3z=0,\quad 3x+ky-2z=0\quad 2x+3y-4z=0$$
Then $$\displaystyle \frac{2k}{11}$$ is equal to


Solution

For non-trivial solution
$$\Delta = \begin{vmatrix} 1&k&3  \\ 3 & k & -2 \\2&3&-4 \end{vmatrix}=0$$
Expanding  along first column,
$$ 1(-4k+6)-3(-4k-9)+2(-2k-3k)=0\Rightarrow -2k+33=0 \Rightarrow k = 33/2$$
 

Mathematics

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