gravitationally bound systems

Imagine you’ve assembled a self-gravitating bound equilibrated system from \(n\) particles of mass $m$, such that the total mass is \(M=nm\) and such that the system has a gravitational binding energy \(E\). Now you reassemble the system from \(p\) times more particles of masses \(m/p\) so that the total mass would be identical, \(M = np \times m/p=nm\). Would the gravitational binding still be \(E\)?

bonus question: Why are the traces \(\mathrm{tr}(A^n)\) of a square matrix \(A\) taken to the power \(n\) invariants? How are they related to the eigenvalues of \(A\)? Would that be a viable way for computing the eigenvalues?