Symmetric And Skew-symmetric
Every square matrix can be written uniquely as the sum of a symmetric matrix and a skew-symmetric matrix; the symmetric part of a square matrix A is correctly given by which expression?
Select the correct option:
Solution
(1/2)(A+AT)
Any square matrix decomposes uniquely into symmetric and skew-symmetric parts, a structural theorem frequently tested in JEE Advanced. A symmetric matrix S satisfies S^T = S, while a skew-symmetric matrix K satisfies K^T = -K. Writing A = S + K and taking the transpose gives A^T = S - K. Adding the two relations, A + A^T = 2S, so S = (1/2)(A + A^T), which indeed satisfies S^T = S. Subtracting gives K = (1/2)(A - A^T). Option (1/2)(A - A^T) is the skew-symmetric part, not the symmetric part. Option A + A^T is symmetric but lacks the normalizing factor 1/2, so it is twice the symmetric part. Option A^T - A is skew-symmetric and negatively scaled. Hence the symmetric part is (1/2)(A + A^T). Plausibility check: transposing (1/2)(A + A^T) gives (1/2)(A^T + A), identical to the original, confirming it is genuinely symmetric and correctly normalized.
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About This Question
- Subject
- mathematics
- Chapter
- matrices and determinants
- Topic
- symmetric and skew-symmetric
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
(1/2)(A+AT)
Any square matrix decomposes uniquely into symmetric and skew-symmetric parts, a structural theorem frequently tested in JEE Advanced. A symmetric matrix S satisfies S^T = S, while a skew-symmetric matrix K satisfies K^T = -K. Writing A = S + K and taking the transpose gives A^T = S - K. Adding the two relations, A + A^T = 2S, so S = (1/2)(A + A^T), which indeed satisfies S^T = S. Subtracting gives K = (1/2)(A - A^T). Option (1/2)(A - A^T) is the skew-symmetric part, not the symmetric part. Option A + A^T is symmetric but lacks the normalizing factor 1/2, so it is twice the symmetric part. Option A^T - A is skew-symmetric and negatively scaled. Hence the symmetric part is (1/2)(A + A^T). Plausibility check: transposing (1/2)(A + A^T) gives (1/2)(A^T + A), identical to the original, confirming it is genuinely symmetric and correctly normalized.
This easy difficulty mathematics question is from the chapter matrices and determinants, covering the topic of symmetric and skew-symmetric. It appeared in the 2025 exam.
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