Asymptotic normality — independence of $\bar X_n$ and $S_n^2$ under normality

Suppose $\{X_1,X_2,\dots,X_n\}$ is a random sample from $N(\mu,\sigma^2)$ distribution. Then the approximate dispersion matrix of $\sqrt n\left((\bar X_n,S_n^2)'-(\mu,\sigma^2)'\right)$ is: