= Solution
Let each observation have dimension $p$, and define the <sample mean> and the divisor-$n$ <sample covariance matrix> by
$$
\bar y=\frac1n\sum_{i=1}^ny_i,\qquad S=\frac1n\sum_{i=1}^n(y_i-\bar y)(y_i-\bar y)^T.
$$
The <multivariate normal distribution> gives the <log-likelihood>
$$
\ell(\mu,V)=-\frac{np}{2}\log(2\pi)-\frac n2\log\det V
-\frac12\sum_i(y_i-\mu)^TV^{-1}(y_i-\mu).
$$
Writing $y_i-\mu=(y_i-\bar y)+(\bar y-\mu)$ makes the cross terms vanish, and the last sum becomes $n\operatorname{tr}(V^{-1}S)+n(\bar y-\mu)^TV^{-1}(\bar y-\mu)$. Since $V$ is a <positive-definite matrix>, its minimum over the mean is at $\widehat\mu=\bar y$. Consequently
$$
\ell_p(V):=\max_\mu\ell(\mu,V)
=-\frac n2\log\det V-\frac n2\operatorname{tr}(V^{-1}S)+C.
$$
When $S$ is a <positive-definite matrix>, $\log\det(V^{-1}S)=\log\det S-\log\det V$. Thus
$$
\ell_p(V)=\frac n2\log\det(V^{-1}S)-\frac n2\operatorname{tr}(V^{-1}S)+C',
$$
where $C'$ depends on the data but not on $V$.
For the <maximum-likelihood estimator>, consider the <positive-definite matrix> $Q=S^{1/2}V^{-1}S^{1/2}$, with positive <eigenvalues> $a_1,\ldots,a_p$. The part of the <profile likelihood> depending on $V$ is $\frac n2\sum_j(\log a_j-a_j)$. The elementary inequality $\log a-a\leq-1$, with equality exactly at $a=1$, shows that the unique maximizing matrix is $Q=I$. Hence
$$
\boxed{\widehat\mu=\bar y,\qquad\widehat V=S.}
$$
The divisor is $n$, not the unbiased covariance divisor $n-1$. The regularity qualification matters: if $S$ is singular, the displayed logarithm of its determinant is not finite, and no positive-definite covariance maximizes the likelihood. Sending the fitted variance in a null direction of $S$ to zero makes the likelihood unbounded. For a nonsingular normal population, $S$ is positive definite almost surely when $n>p$, which is the setting of the large-sample test.
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