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If A3x3 is a matrix with |A| = 2. What is the determinant of Adj (Adj (Adj A))?

asked in Linear Algebra by Active (2.4k points) | 1.1k views

 

|Adj (Adj (Adj A)) | = |A| ((n-1)^3)

|A|=2

|Adj (Adj (Adj A)) |  = 2^8 =256

1 Answer

+8 votes
Best answer

We know :

| Adj(A) |  = |A|n-1

Now extending this result we can solve as :

          Adj(A) . Adj(Adj(A))  = |Adj(A)| . I where I is identity matrix

 ==>  | Adj(A) . Adj(Adj(A)) |  =  |Adj(A)|n

 ==>  | Adj(Adj(A)) |          =   |Adj(A)|n-1   = |A|(n-1)^2

 ==> | Adj(Adj(A)) * Adj( Adj(Adj(A)))|    =  | Adj(Adj(A)) |n

 ==> | Adj( Adj(Adj(A))) |      =  | Adj(Adj(A)) |n-1

 ==>  | Adj( Adj(Adj(A))) |     =  | A |(n-1)^3

Now n = 3 here and given |A|  = 2

Therefore 

          | Adj( Adj(Adj(A))) |    = 22^3

==>    | Adj( Adj(Adj(A))) |    = 256

Hence 256 is the correct answer..

answered by Veteran (96.6k points)
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