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For how many values of k do the homogeneous equations 4x + ky + 2z = 0, kx + 4y + z = 0, and 2x + 2y + z = 0 admit a non-trivial solution?

1. 2
2. 1
3. zero
4. 3

Correct answer: 2

Solution

The homogeneous system of equations admits a non-trivial solution when the determinant of the coefficient matrix is zero. By calculating the determinant and setting it to zero, we find that there are two specific values of k that satisfy this condition.

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