[Math] Determinant of a N symmetric square matrix with diagonal 1

Question

What is the determinant of a symmetric $n \times n$ matrix with all diagonals be 1 and all others are $\rho$ (yes correlation matrix)?

Anyone can tell me a method to work it out elegantly?

Thanks!

Answer 1

Let $A$ be an $n\times n$ matrix of your desired form. Let us form the matrix of all ones, call it $J$. Then we can represent $A$ as $$A=\rho J - (\rho - 1)I$$ The determinant of the above matrix is $$\det(A) = \rho^n\det\left(J - \frac{\rho - 1}{\rho}I\right)$$ Letting $\lambda = \frac{\rho - 1}{\rho}$, the latter determinant is precisely the characteristic polynomial of $J$, easily seen as $$p(\lambda)=(-1)^{n}\lambda^{n-1}(\lambda - n)$$ Some simplifying then gives the determinant as $$\det(A)=\left(1-\rho \right)^{n-1}\left(1 + \rho n-\rho\right)$$