![Q. No9. For a Matrix M = [mij]; i,j =1,2,3,4, diagonal elements are all zero and mij = -mji The minimum number of elements required to fully specify the matrix __ A. 0 B. 6 C. 12 D. 16](https://www.letstalkacademy.com/wp-content/uploads/2026/01/slide_9.png)
Q. No9. For a Matrix M = [mij]; i,j =1,2,3,4, diagonal elements are all zero and mij = –mji
The minimum number of elements required to fully specify the matrix __
A. 0
B. 6
C. 12
D. 16
The minimum number of elements required to fully specify the 4×4 matrix M with zero diagonal elements and mij=−mji is 6.
This condition defines a skew-symmetric matrix, where specifying the upper triangular elements determines the rest.
Matrix Properties
A 4×4 matrix has 16 total elements. The diagonal elements mii must be zero for i=1,2,3,4, leaving 12 off-diagonal elements.
The antisymmetric condition mij=−mji pairs each upper triangular element with its negative in the lower triangle, making only 6 elements independent.
Visual Structure
Consider the matrix form:
M = [
0 a b c
−a 0 d e
−b −d 0 f
−c −e −f 0
]
Here, a,b,c,d,e,f are the 6 free parameters that fully specify M.
Option Analysis
- A. 0: Incorrect, as off-diagonal values can vary freely.
- B. 6: Correct, matching the upper triangular independent entries in a 4×4 skew-symmetric matrix.
- C. 12: Counts all off-diagonals but ignores pairing via antisymmetry.
- D. 16: Total entries without constraints.
Minimum Elements to Specify Skew-Symmetric 4×4 Matrix with Zero Diagonal
Skew-symmetric matrices with zero diagonal elements and the property mij=−mji are key in linear algebra, especially for CSIR NET Life Sciences and GATE exams. This article solves the exact question: For matrix M = [mij]; i,j=1,2,3,4, diagonal elements zero and mij = -mji, the minimum number of elements to fully specify it (options 0,6,12,16).
What is a Skew-Symmetric Matrix?
A skew-symmetric matrix satisfies MT=−M, implying zero diagonals (mii=−mii⇒mii=0) and antisymmetry off-diagonal. For n=4, total entries=16, fixed zeros=4, paired off-diagonals=12, independent upper triangle entries = 4(4−1)/2=6.
Step-by-Step Derivation
- Total positions: 4×4=16.
- Diagonals fixed at 0: 4 positions.
- Off-diagonals form 6 pairs (e.g., m12=-m21, m13=-m31).
- Specify 6 values above diagonal to fill all.
| Property | Count for 4×4 |
|---|---|
| Total elements | 16 |
| Zero diagonals | 4 |
| Independent elements | 6 |
| Paired off-diagonals | 12 |
Why 6 for CSIR NET/GATE?
This tests degrees of freedom in matrices. General formula for nxn skew-symmetric: n(n−1)/2. For n=4: 6. Matches exam patterns in quantitative aptitude.
Common Mistakes
- Choosing 12 ignores antisymmetry.
- 16 forgets constraints.
- 0 assumes all fixed (only diagonals are).
Practice with examples: Set a=1,b=2,c=3,d=4,e=5,f=6 to generate full M. Verify MT=−M.
