Matrix Properties
Hardmathematics
If A is a skew-symmetric matrix of odd order, then |A| equals
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Solution
Incorrect! Answer:
0
- Definition: For a skew-symmetric matrix A, AT=−A.
- Apply Determinant: ∣AT∣=∣−A∣.
- Properties:
- ∣AT∣=∣A∣
- ∣−A∣=(−1)n∣A∣, where n is the order.
- Constraint of Odd Order: If n is odd, (−1)n=−1.
- Thus, ∣A∣=−∣A∣⟹2∣A∣=0⟹∣A∣=0.
- Result: The determinant of an odd-order skew-symmetric matrix is always zero.
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About This Question
- Subject
- mathematics
- Chapter
- matrices and determinants
- Topic
- matrix properties
- Difficulty
- Hard
- Year
- 2025
Solution
Correct Answer:
0
- Definition: For a skew-symmetric matrix A, AT=−A.
- Apply Determinant: ∣AT∣=∣−A∣.
- Properties:
- ∣AT∣=∣A∣
- ∣−A∣=(−1)n∣A∣, where n is the order.
- Constraint of Odd Order: If n is odd, (−1)n=−1.
- Thus, ∣A∣=−∣A∣⟹2∣A∣=0⟹∣A∣=0.
- Result: The determinant of an odd-order skew-symmetric matrix is always zero.
This hard difficulty mathematics question is from the chapter matrices and determinants, covering the topic of matrix properties. It appeared in the 2025 exam.
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