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Question

Let A be a 3×3 real matrix. If det(2 Adj(2 Adj(Adj(2A))))=241, then the value of det(A2) equals

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Solution

We know that, if A is a square matrix of order n, then
|Adj A|=|A|n1
Adj(Adj A)=|A|n2A
|kA|=kn|A| where k is a scalar
Adj (kA)=kn1Adj (A)

det(2 Adj(2 Adj(Adj(2A))))=241
det(2 Adj(2 Adj(22Adj(A))))=241
det(2Adj(224Adj(Adj(A))))=241
det(2Adj(25|A|A))=241
det(2(32 |A|)2(Adj(A))=241
det(211|A|2(Adj(A))=241
(211|A|2)3|AdjA|=241
233|A|6 |A|2=241
|A|8=28|A|=±2
|A|2=4

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