GATE BT 2024 • Q28
MCQ • 1 Mark
Section 1: Engineering Mathematics
Linear Algebra
Subtopic: Fundamentals of Linear Algebra
Question Statement:
A value of k for which the linear equations (k - 1)x + 3y = 0 and 2x + ky = 0
have a non-zero solution is ________.
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2024 Q.28)**
**1. Governing Concept**:
- **Topic**: Linear Algebra & Homogeneous Linear Systems
- The system of linear homogeneous equations:
$\begin{cases} (k - 1)x + 3y = 0 \\ 2x + ky = 0 \end{cases}$
- For a non-trivial (non-zero) solution to exist, the coefficient matrix determinant must be zero:
$\begin{vmatrix} k - 1 & 3 \\ 2 & k \end{vmatrix} = 0 \implies k(k - 1) - (2)(3) = 0$
$k^2 - k - 6 = 0 \implies (k - 3)(k + 2) = 0 \implies k = 3 \text{ or } k = -2$
**2. Option Analysis**:
- **Option (A) [1] is INCORRECT**: Determinant equals $1(0) - 6 = -6 \neq 0$.
- **Option (B) [2] is INCORRECT**: Determinant equals $2(1) - 6 = -4 \neq 0$.
- **Option (C) [3] is CORRECT**: For $k = 3$, determinant $= 3(2) - 6 = 0$, yielding non-trivial solutions.
- **Option (D) [4] is INCORRECT**: Determinant equals $4(3) - 6 = 6 \neq 0$.
**3. Final Answer**:
- **Correct Choice**: **C**
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