Inverse of a symmetric matrix

Question

Let we have a symmetric matrix

$\Sigma=(1-\rho)I_p+\rho\mathbf1\mathbf1'$

I am reading a textbook which says that

$\Sigma^{-1}=\frac{-\rho}{(1-\rho)(p\rho+1-\rho)}\mathbf1\mathbf1'+\frac{1}{1-\rho}I_p$

Could you please share some pointer if there is any rule/theorem to arrive this expression of the inverse?

Answer 1

You might want to look at the Sherman-Morrison formula, matrix inversion lemma