![Q.9 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 (A) 0 (B) 6 (C) 12 (D) 16](https://www.letstalkacademy.com/wp-content/uploads/2026/01/slide_9.jpg)
Q.9 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
(A) 0
(B) 6
(C) 12
(D) 16
Minimum Elements to Specify 4×4 Skew-Symmetric Matrix
The matrix M is a 4×4 skew-symmetric matrix where all diagonal elements are zero and off-diagonal elements satisfy mij = -mji. This structure means only the upper or lower triangular independent elements need specification. The correct answer is 6, as detailed below.
Matrix Properties
A skew-symmetric matrix follows MT = -M, forcing mii = 0 for i=1 to 4 and mij = -mji for i ≠ j. In a 4×4 matrix, 16 total positions exist, but 4 diagonals are fixed at zero. The 12 off-diagonals form 6 symmetric pairs (e.g., m12 determines m21 = -m12), leaving 6 independent values.
Visual Example
Consider this 4×4 skew-symmetric form:
| 0 | a | b | c |
| -a | 0 | d | e |
| -b | -d | 0 | f |
| -c | -e | -f | 0 |
Here, specifying a, b, c, d, e, f (6 elements) fully defines M.
Option Analysis
GATE Relevance
This question tests matrix symmetry properties, common in exams like GATE Chemical/Engineering Maths. For n×n skew-symmetric matrices, independent elements are n(n-1)/2; for n=4, that’s 6.
