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For the system x-ky+z=0 kx+3y-kz=0 3x+y-z=0 if the zero solution is the only solution, then the possible values of k are:

1. R∖{2,-3}
2. R∖{2}
3. R∖{-3}
4. {2,-3}

Correct answer: R∖{2,-3}

Solution

The zero solution being the only solution indicates that the system of equations must be consistent and have a unique solution, which occurs when the determinant of the coefficient matrix is non-zero. The values of k that make the determinant zero, specifically k = 2 and k = -3, must be excluded, leading to the conclusion that k can take any real number except these two.

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