For a matrix M = [mij]; i,j = 1,2,3,4, the diagonal elements are all zero and mij = -mji. The minimum number of elements required to fully specify the matrix is:
Options:
(A) 0
(B) 6
(C) 12
(D) 16
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2020 Q.9)**
**1. Governing Concept**:
- **Topic**: Linear Algebra & Skew-Symmetric Matrices
- A matrix $M = [m_{ij}]$ of order $4 \times 4$ with $m_{ii} = 0$ and $m_{ij} = -m_{ji}$ is a **skew-symmetric matrix**.
- In any $n \times n$ skew-symmetric matrix:
- All $n$ main diagonal elements are identically zero ($m_{ii} = 0$, requiring 0 parameters).
- The $n(n - 1)/2$ elements below the main diagonal are completely determined by the elements above the diagonal ($m_{ji} = -m_{ij}$).
- Therefore, the number of independent elements required to fully specify the matrix is:
$\text{Independent elements} = \frac{n(n - 1)}{2} = \frac{4(4 - 1)}{2} = \frac{12}{2} = 6$
**2. Option Analysis**:
- **Option (A) [0] is INCORRECT**: Zero elements would mean a fixed null matrix.
- **Option (B) [6] is CORRECT**: Exactly $\frac{4 \times 3}{2} = 6$ strictly independent off-diagonal upper triangular elements.
- **Option (C) [12] is INCORRECT**: Total number of non-diagonal elements ($16 - 4 = 12$), ignoring skew-symmetry.
- **Option (D) [16] is INCORRECT**: Total elements in an unconstrained $4 \times 4$ matrix.
**3. Final Answer**:
- **Correct Choice**: **B**
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