Q.30 If ܣ and ܤ are two skew-symmetric matrices, the matrix ܣܣܤ ܤ must be
(A) skew-symmetric (B) symmetric
(C) invertible (D) NOT invertible
The matrix AB+BA is symmetric when A and B are skew-symmetric matrices. This follows from the properties of matrix transposes, making option (B) correct for this CSIR NET-level question.
Problem Analysis
Skew-symmetric matrices satisfy AT=−A and BT=−B. The expression AB+BA requires checking its transpose to classify it.
Proof of Symmetry
Consider (AB+BA)T=(AB)T+(BA)T=BTAT+ATBT. Substituting skew-symmetric properties gives BTAT+ATBT=(−B)(−A)+(−A)(−B)=BA+AB. Thus, (AB+BA)T=AB+BA, proving symmetry.
Option Evaluation
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(A) Skew-symmetric: Incorrect, as (AB+BA)T=AB+BA≠−(AB+BA).
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(B) Symmetric: Correct, per the proof above.
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(C) Invertible: Incorrect; no guarantee of non-zero determinant (e.g., zero matrix case).
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(D) NOT invertible: Incorrect; can be invertible depending on A and B.
Introduction: Skew Symmetric Matrices AB + BA Symmetric Property
In linear algebra for CSIR NET exams, understanding skew symmetric matrices AB + BA symmetric property is crucial. When A and B are skew-symmetric (AT=−A, BT=−B), the matrix AB+BA proves symmetric, a key result for competitive exams.
Skew-Symmetric Matrix Definition
A square matrix K is skew-symmetric if KT=−K, with zero diagonal elements. Sum of skew-symmetric matrices remains skew-symmetric, but products yield different forms.
Detailed Proof: Why AB + BA is Symmetric
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Start: (AB+BA)T=(AB)T+(BA)T.
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Transpose rule: =BTAT+ATBT.
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Skew-symmetry: =(−B)(−A)+(−A)(−B)=BA+AB.
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Conclusion: Equals original, so symmetric.
CSIR NET Exam Relevance
This property appears in matrix theory questions testing transpose operations. Related: For symmetric A, B, AB−BA is skew-symmetric.
Applications and Examples
In physics, skew-symmetric matrices model rotations; AB+BA symmetry aids quadratic forms. Example: 2×2 skew-symmetric yields symmetric anticommutator.
