# How can I calculate $\frac{\partial \log |\Sigma|}{\partial \rho }$ where $\Sigma=(1-\rho)I+\rho\mathbf{1}\mathbf{1}^\top$?

I need to calculate the $$\dfrac{\partial \log |\Sigma|}{\partial \rho }$$ when $$\Sigma = (1-\rho) I + \rho \mathbf{1} \mathbf{1}^\top$$ and $$\Sigma$$ has dimension $$p \times p$$.

I try to use the formula presented here and here but the result is not right.

• Without trying anything fancy, you can simply calculate $|\Sigma|$ and proceed as usual. – StubbornAtom Apr 25 at 15:38

The derivative wrt $$\rho$$ is $$\operatorname{trace}((-I+11^T){\Sigma}^{-1})$$.