GATE BT 2024 • Q62
NAT • 2 Marks
Section 1: Engineering Mathematics
Linear Algebra
Subtopic: Fundamentals of Linear Algebra
Question Statement:
If v = (
2
2
2
) is an eigenvector of the matrix (
1 2 3
1 2 3
1 2 3
) corresponding to the
non-zero eigenvalue, 𝜆, then the value of 𝜆 is ________.
Official Answer Key:
6 to 6
Step-by-Step Solution & Explanation
### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2024 Q.62)**
**1. Governing Concept & Definition**:
- **Topic**: Linear Algebra & Eigenvalue-Eigenvector Relationship
- By definition of an eigenvector $v$ and eigenvalue $\lambda$:
$A v = \lambda v$
**2. Detailed Step-by-Step Calculation**:
- **Given Matrix & Vector**:
$A = egin{pmatrix} 1 & 2 & 3 \ 1 & 2 & 3 \ 1 & 2 & 3 \end{pmatrix}, \quad v = egin{pmatrix} 2 \ 2 \ 2 \end{pmatrix}$
- **Matrix-Vector Multiplication**:
$A v = egin{pmatrix} 1(2) + 2(2) + 3(2) \ 1(2) + 2(2) + 3(2) \ 1(2) + 2(2) + 3(2) \end{pmatrix} = egin{pmatrix} 2 + 4 + 6 \ 2 + 4 + 6 \ 2 + 4 + 6 \end{pmatrix} = egin{pmatrix} 12 \ 12 \ 12 \end{pmatrix}$
- **Finding Eigenvalue $\lambda$**:
$egin{pmatrix} 12 \ 12 \ 12 \end{pmatrix} = \lambda egin{pmatrix} 2 \ 2 \ 2 \end{pmatrix} \implies 2\lambda = 12 \implies \lambda = 6$
**3. Final Answer**:
- **Calculated Value**: **6**
- **Official GATE Accepted Range**: **6 to 6**
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