Discussion
If the system of linear equations \(x + 2y - z = 0,\ 3x + (k+7)y - 3z = 0,\ 2x + 4y + (k-3)z = 0\) has a non-trivial solution, then the value of \(k\) is:
Answer: B
A homogeneous system has a non-trivial solution only when the determinant of its coefficient matrix is zero:
\[ \begin{aligned} \begin{vmatrix} 1 & 2 & -1 \\ 3 & k+7 & -3 \\ 2 & 4 & k-3 \end{vmatrix} \\ &= 0 \end{aligned} \]Expanding along the first row: \((k^2 + 4k - 9) - 2(3k - 3) - (12 - 2k - 14) = k^2 - 1\). So \(k^2 = 1\) and \(k = \pm 1\). The options -1 and 1 each give only one of the two roots, and k = 0 gives determinant -1, so the system then has only the trivial solution.
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