Finding the eigenvalues of a matrix

Let A be a square matrix of real numbers whose eigenvalues are positive integers.

It is given that |adj(adj(A))| = 81.

What is the characteristic polynomial of the matrix?

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What I did was this:

|adj(adj(A))| = |adj(A)|^(n-1) = |A|^(n-1)^2 = 81

But I don't know how to proceed.

Any suggestions? Or if there's any other possible way...

Thanks!

Re: Finding the eigenvalues of a matrix

The determinant of A^{n-1} is either -9 or 9.

Re: Finding the eigenvalues of a matrix

Re: Finding the eigenvalues of a matrix