Determinant and Inverse of a predefined matrix.
See the attachment. where (a, b) = gcd(a, b), mu = Mobius functions, phi = Euler function
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What are mu and phi?
Mobius functions, Euler function
One needs to spend time analyzing the question for the first few values of n to develop an intuition and determine if a general solution exists. I would say this is a very time-consuming and challenging question that deserves a higher bounty.
OK, I will increase bounty.