InterviewSolution
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InterviewSolution
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Show that the matrix B′ AB is symmetric or skew-symmetric according as A is symmetric or skew-symmetric. |
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Answer» We suppose that A is a symmetric matrix, then A′ = A Consider (B′ AB)′ ={B′ (AB)}′ = (AB)′ (B′ )′ [∵ (AB)′ = B ′ A′ ] = B′ A′ (B) [∵ (B′ )′ = B] = B′ (A′ B) = B′ (AB) [∵ A′ = A] ⇒ (B′ AB)′ = B′ AB which shows that B′ AB is a symmetric matrix. Now, we suppose that A is a skew-symmetric matrix. Then, A′ = − A Consider (B′ AB)′ = [B′ (AB)]′ = (AB)′ (B′ )′ [∵(AB)′ = B′ A′ and (A′ )′ = A] = (B′ A′ )B = B′ (−A)B= − B′ AB [∵ A′ = − A] ⇒ (B′ AB)′ = − B′ AB which shows that B′ AB is a skew-symmetric matrix. |
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