For what value of parameter $k$ does the homogeneous linear system $$\begin{aligned} x + 2y - z &= 0 \\ 2x + 5y + z &= 0 \\ 3x + 7y + k z &= 0 \end{aligned}$$ have non-trivial (non-zero) solutions?

(A) 0

(B) 1

(C) -1

(D) 2

Answer Key:

A

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Condition for Non-Trivial Solution:** $\det(A) = 0$. 2. **Stage 2: Determinant Expansion:** $\det(A) = 1(5k - 7) - 2(2k - 3) - 1(14 - 15) = (5k - 7) - (4k - 6) - (-1) = k - 7 + 6 + 1 = k$. 3. **Stage 3: Equate to Zero:** $k = 0$. 4. **Stage 4: Selection:** Option A is correct.

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