matrix

Question

If A_3x3 is a matrix with |A| = 2. What is the determinant of Adj (Adj (Adj A))?

Comments

|Adj (Adj (Adj A)) | = |A| ^{((n-1)^3)}

|A|=2

|Adj (Adj (Adj A)) |  = 2^8 =256

Answer 1

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))) |    = 2^{2^3}

==>    | Adj( Adj(Adj(A))) |    = 256

Hence 256 is the correct answer..