Be $\Sigma = (1-\rho) I_p + \rho 1_p 1_p^\top$. I need to write $\rho$ in function of $\Sigma$ to calculate the maximum likelihood estimator of $\rho$ and I know that the maximum likelihood estimator of $\Sigma$ in the general case (when we have a independent random sample of a multivariated normal distribution with mean $\mu$ and covariance matrix $\Sigma$) is $S$. In this case, I have a independent random sample of a multivariated normal distribution with mean $0$ and covariance matrix $ \Sigma = (1-\rho) I_p + \rho 1_p 1_p^\top$).
I want to use the invariance property of a mle. I try to write $\rho$ in function of a determinant of $\Sigma$ but I had an incorrect result.