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Q.

Let B be a skew symmetric matrix of order  3×3  with real entries.
Given I – B and I + B are non–singular matrices.
If  A=(I+B)(IB)1  where det(A) > 0, then the value of det(2A) – det(adjA) is
[Here det(P) denotes determinant of square matrix P and det(adj P) denotes determinant of adjoint of square matrix P respectively]

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answer is 7.

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Detailed Solution

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A=(I+B)(IB)1       AAT=(I+B)(IB)1((IB)T)1(I+B)T =(I+B)(IB)1(I+B)1(IB)     =(I+B)((IB)(IB))1(IB)  =(I+B)((IB)(I+B))1(IB)

( I + B, I  B are commutative)

AAT=(I+B)(I+B)1(IB)1(IB)=I

AAT=I

|AA|T=|I|=1|A|2=1  as  |A|>0|A|=1

det (2A) – det(adj(A)) = 8 det(A) –  det(A)2  =8(1)(1)2=7

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Let B be a skew symmetric matrix of order  3×3  with real entries.Given I – B and I + B are non–singular matrices.If  A=(I+B)(I−B)−1  where det(A) > 0, then the value of det(2A) – det(adjA) is[Here det(P) denotes determinant of square matrix P and det(adj P) denotes determinant of adjoint of square matrix P respectively]