Question

If A is a skew-symmetric matrix of order n. then

• A. \(a_{ij} = \frac{1}{a_{ij}}\), for all values of i and j.
• B. \(a_{ij} = 0\), where i = j
• C. \(a_{ij} \neq 0\), for all values of i and j.
• D. \(a_{ij} \neq 0\), where i = j only.

Answer 1

The Correct Option is B

The solution to understanding a skew-symmetric matrix and identifying the correct property involves recognizing key characteristics of such matrices.

1. Definition: A square matrix \( A \) of order \( n \) is called skew-symmetric if its transpose \( A^T \) equals the negative of the matrix itself, i.e., \( A^T = -A \).
2. Elements Property: This implies that for all elements \( a_{ij} \) of the matrix:

If \( A = [a_{ij}] \), then:

• \( a_{ij} = -a_{ji} \) for all \( i \) and \( j \).
• \( a_{ii} = -a_{ii} \) means \( 2a_{ii} = 0 \), thus \( a_{ii} = 0 \).

3. Conclusion: Consequently, in a skew-symmetric matrix, all diagonal elements must be zero, i.e., \( a_{ij} = 0 \) where \( i = j \).

Property	Expression
Skew-symmetric	\( a_{ij} = -a_{ji} \)
Diagonal	\( a_{ii} = 0 \)

Hence, the correct answer is that in a skew-symmetric matrix, \( a_{ij} = 0 \) where \( i = j \).