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Question

Let A be a square matrix of order 3 whose elements are real numbers and adj (adj (adj A))=1603040034. Then the absolute value of trace(A1) is

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Solution

adj (adj A)=|A|n2A
So, adj (adj adj A)=|adj A|n2adj A
=(|A|n1)n2 adj A
Putting n=3, we get
adj (adj adj A)=|A|2adj A=1603040034 (1)
Taking determinant both sides
|adj (adj adj A)|=|A|6|adj A|=256
|adj (adj adj A)|=|A|6|A|2=28
|A|=±2

From (1),
adj A=141603040034=⎢ ⎢ ⎢ ⎢ ⎢40340100341⎥ ⎥ ⎥ ⎥ ⎥

A1=adj A|A|=±⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢2038012003812⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥|trace(A1)|=3


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