Fisher information of a multivariate normal location model ($I(\theta )={\Sigma }^{-1}$)

For $X\sim N_p(\theta ,\Sigma )$ with known nonsingular covariance matrix, the score function is ${\Sigma }^{-1}(X-\theta )$ and the per-observation Fisher information matrix is ${\Sigma }^{-1}$. In particular, covariance $I_p$ gives information $I_p$.