Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.11)**
**1. Intuitive Physical Setup & Given Parameters**:
We are asked to identify the defining mathematical property for the transpose of any real symmetric matrix $A$.
**2. Step-by-Step Mathematical Derivation**:
1. **Mathematical Definition of a Symmetric Matrix**:
A square matrix $A \in \mathbb{R}^{n \times n}$ is defined as symmetric if and only if each element satisfies:
a_{ij} = a_{ji} \quad \forall i, j \in \{1, 2, \dots, n\}
2. **Transpose Operation**:
The transpose $A^T$ is formed by swapping rows and columns ($[A^T]_{ij} = a_{ji}$).
Since $a_{ji} = a_{ij}$, it follows immediately by definition that:
A^T = A
3. **Contrast with Skew-Symmetric & Orthogonal Matrices**:
- Skew-symmetric: $A^T = -A$
- Orthogonal: $A^T = A^{-1}$
**3. Comprehensive Options & Distractor Analysis**:
- **(A) inverse of A**: ❌ **INCORRECT**. This holds for orthogonal matrices ($A^T = A^{-1}$), not general symmetric matrices.
- **(B) null matrix**: ❌ **INCORRECT**. Only true if $A = 0$.
- **(C) -A**: ❌ **INCORRECT**. This defines a skew-symmetric (anti-symmetric) matrix.
- **(D) A**: ✅ **CORRECT**. By definition, a real symmetric matrix is self-transpose ($A^T = A$).
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **D**
- **Exam Takeaway**: Real symmetric matrices have three fundamental properties in GATE: 1) $A^T = A$, 2) All eigenvalues are strictly real, 3) Eigenvectors corresponding to distinct eigenvalues are mutually orthogonal.