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NCERT Class 12 · Ch 3 Miscellaneous Exercise · Q2

NCERT Class 12 Maths · Chapter 3 Matrices · Miscellaneous Exercise Question 2

Show BᵀAB is symmetric or skew symmetric according as A is symmetric or skew symmetric. Assume compatible dimensions.

💡 Concept Hints

  1. Check the matrix orders and the meaning of the requested operation before calculating.
  2. Keep row and column indices in order. Compare corresponding entries only after the matrix expressions have compatible dimensions.

✅ Step-by-Step Solution

Final Answer: BᵀAB inherits the same transpose sign as A.
  1. Let Aᵀ=εA, where ε=1 in the symmetric case and ε=−1 in the skew-symmetric case.
  2. (BᵀAB)ᵀ=BᵀAᵀ(Bᵀ)ᵀ=Bᵀ(εA)B=εBᵀAB.
  3. Thus its transpose equals itself or its negative, respectively. It is square because BᵀAB has order equal to the column count of B.

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