For what value of K do the following system of equations possess a non trivial (i.e. not all zero) solution over the set of rationals Q ?
`x + K y + 3 z = 0 , 3 x + K y - 2 z = 0` ,
`2 x + 3 y - 4 z = 0`.
For that value of K , find all the solutions of the system.

Explanation

To find the value of \( K \) for which the system of equations has a non-trivial solution, we need to set up the system of equations and find the determinant of the coefficient matrix. The equations given are:

1. \( x + K y + 3 z = 0 \)
2. \( 3 x + K y - 2 z = 0 \)
3. \( 2 x + 3 y - 4 z = 0 \)

We can express this system in matrix form as follows:

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