Matrix doubt

Question

If matrix A =  

1	2	-1
-1	1	2
2	-1	1

Then find det(adj(adj A)) is

a) 14^4

b)14^6

c)14^9

d)14^2

Comments

$14^{4}$

Use $|adj(adjA)|={|A|^{(n-1)}}^{2}$

Answer 1

We know that

A.adj(A)=adj(A).A=|A|.I

http://mathinstructor.net/2012/04/how-to-prove-that-a-adja-adja-adeta-i/

Take the determinant both side , it becomes

|A.adj(A)| = | |A|.I |                 // |KD|=Kⁿ.|D|   ; K is constant and D is n*n square matrix

so |A.adj(A)| = |A|ⁿ.|I|            //  |I|=1

=> |A|.|adj(A)|=|A|ⁿ             //   |A.B|=|A||B|

=> |adj(A)|=|A|^n-1   

Now adjoint of adjoint of A

| adj(adj(A)) | =?

Let adj(A)=B        

so | adj(adj(A)) | =|adj(B)|       

|adj(B)|  = |B|^n-1      //   |adj(A)|=|A|^n-1   

Put the value of B and get | adj(adj(A)) |   = |adj A|^n-1   = (|A|^n-1 )^n-1 = |A|^(n-1)*(n-1)

| adj(adj(A)) |  =|A|^(n-1)(n-1)