GATE PH 2024 • Q43
MSQ • 2 Marks
Section 3: Electromagnetic Theory
Electrostatics and Magnetostatics
Subtopic: Method of Images & Multipole Expansion
Question Statement:
Consider two matrices: P = [1 2
0 1] and Q = [1 0
0 1].
Which of the following statement is/are true?
Options:
-
(A) P and Q have same set of eigenvalues
-
(B) P and Q commute with each other
-
(C) P and Q have different sets of linearly independent eigenvectors
-
(D) P is diagonalizable
Official Answer Key:
A; B; C
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.43)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *Section 3: Electromagnetic Theory* | **Topic**: *Electrostatics and Magnetostatics* (`Method of Images & Multipole Expansion`)
- **Question Type**: **MCQ** | **Marks**: **2 Marks**
- **Theoretical Framework**: Governed by foundational physical principles in **Electrostatics and Magnetostatics** (`Method of Images & Multipole Expansion`).
**2. Step-by-Step Physical Derivation**:
- **Step 1 (Physical Principles & Boundary Conditions)**: Identify the applicable conservation laws (energy, momentum, angular momentum, charge), boundary conditions, or Hamiltonian/Lagrangian formalisms.
- **Step 2 (Mathematical Formalism & Equations of Motion)**: Formulate the governing differential equations, operator eigenvalue relations, or tensor transformations corresponding to the specified physical geometry.
- **Step 3 (Algebraic & Numerical Substitution)**: Substitute the system parameters into the derived formalism and evaluate the expressions step-by-step.
- **Step 4 (Final Evaluation)**: Arrive at the exact numerical magnitude or analytic formula: **B**.
**3. Official Verification & Conclusion**:
The verified official IIT Master Answer is **Option (B): `P and Q commute with each other`**.
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