= Solution
Use the usual <sample mean> and the divisor-$n-1$ <sample covariance matrix>,
$$
\bar x=\frac1n\sum_{j=1}^nx_j,\qquad S=\frac1{n-1}\sum_{j=1}^n(x_j-\bar x)(x_j-\bar x)^T.
$$
The residual sum is zero, so expanding $x_j-\mu=(x_j-\bar x)+(\bar x-\mu)$ gives
$$
\sum_j(x_j-\mu)(x_j-\mu)^T=(n-1)S+n(\bar x-\mu)(\bar x-\mu)^T.
$$
Multiplying the independent <multivariate normal densities> and taking logarithms yields the <log-likelihood>
$$
\boxed{\ell(\mu,V)=-\frac{np}{2}\log(2\pi)-\frac n2\log|V|-\frac{n-1}{2}\operatorname{tr}(V^{-1}S)-\frac n2(\bar x-\mu)^TV^{-1}(\bar x-\mu).}
$$
Here $|V|$ is the <determinant> and $\operatorname{tr}$ denotes the <trace>. The <maximum-likelihood estimators> are
$$
\boxed{\widehat\mu=\bar x,\qquad \widehat V=\frac{n-1}{n}S.}
$$
If $S$ instead denotes the divisor-$n$ <sample covariance matrix>, replace $(n-1)S$ by $nS$ and then $\widehat V=S$. These formulas describe a nonsingular <maximum-likelihood estimator> when the empirical scatter is <positive-definite>; if it is singular, there is no maximizer over positive-definite <covariance matrices>.
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