Derive lower bound for equicorrelation rho

Let $X, Y, Z$ be zero-mean, unit-variance random variables whose pairwise correlations are all equal to $\rho$. Find the tight lower bound on $\rho$ such that the $3 \times 3$ correlation matrix is positive semidefinite. Show your work by analyzing eigenvalues of the equicorrelation matrix. Then generalize: for $n$ variables with common off-diagonal correlation $\rho$, derive the feasible interval of $\rho$ as a function of $n$.

Overview: This question evaluates understanding of positive semidefiniteness for equicorrelation matrices, eigenvalue analysis, and parameter constraints in multivariate statistics.