Introduction
The given set of equations represents a linear homogeneous system of the form AX = 0 in three variables (x, y, z). A homogeneous linear system is inherently consistent because it unconditionally admits at least the trivial solution (x = y = z = 0). The nature of the solution set—whether it contains only the trivial solution or infinitely many non-trivial solutions—depends directly on the determinant and rank of the coefficient matrix A.
1. Matrix Formulation of the System
The system can be expressed in matrix form as AX = 0, where:
$$A = \begin{bmatrix} 2 & 3k & 3k+4 \\ 1 & k+4 & 4k+2 \\ 1 & k+1 & 3k+4 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad 0 = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$$
2. Row Reduction and Determinant Calculation
To evaluate the determinant |A| efficiently, elementary row operations are performed:
- Interchanging Rows: Swap R1 and R3 (R1 ↔ R3):
$$A \sim \begin{bmatrix} 1 & k+1 & 3k+4 \\ 1 & k+4 & 4k+2 \\ 2 & 3k & 3k+4 \end{bmatrix}$$ - Row Eliminations: Apply R2 → R2 - R1 and R3 → R3 - 2R1:
$$A \sim \begin{bmatrix} 1 & k+1 & 3k+4 \\ 0 & 3 & k-2 \\ 0 & k-2 & -3k-4 \end{bmatrix}$$
Expanding the determinant along the first column:
$$|A| = 1 \cdot [3(-3k - 4) - (k - 2)^2]$$
$$|A| = -9k - 12 - (k^2 - 4k + 4) = -(k^2 + 5k + 16)$$
3. Analysis of Real Roots for k
For a homogeneous system, non-trivial solutions exist if and only if |A| = 0. Setting the quadratic factor to zero:
$$k^2 + 5k + 16 = 0$$
The discriminant (Δ) is given by:
$$\Delta = b^2 - 4ac = 5^2 - 4(1)(16) = 25 - 64 = -39$$
Because Δ < 0, the equation has no real roots. Consequently, |A| ≠ 0 for all real values of k.
4. Determination of Solution Set
- Condition for Non-Trivial Solution: Requires |A| = 0, which is impossible for any real number k.
- Condition for Trivial Solution: When |A| ≠ 0, Rank(A) = 3 = number of unknowns. Thus, the system possesses a unique trivial solution (x = 0, y = 0, z = 0).
Conclusion
Since |A| ≠ 0 for every real number k, the system never possesses non-trivial solutions. However, because every homogeneous system admits the trivial solution x = y = z = 0, the given system has a solution for all real values of k (k ∈ ℝ).