= Solution
Write $Z_i=(X_i,Y_i)^T$ for the independent observations and put $\Omega=\Sigma^{-1}$. Apart from constants, the log-likelihood is
$$
\ell(\Omega)=\frac n2\log|\Omega|
-\frac12\sum_{i=1}^n Z_i^T\Omega Z_i.
$$
Using the stated matrix derivatives, its <score function> is
$$
\frac{\partial\ell}{\partial\Omega}
=\frac n2\Omega^{-1}
-\frac12\sum_{i=1}^nZ_iZ_i^T.
$$
At a root of the score, $\Omega^{-1}=n^{-1}\sum_iZ_iZ_i^T$. Hence the <maximum-likelihood covariance estimator for centered Gaussian data> is
$$
\boxed{
\widehat\Sigma=\frac1n\sum_{i=1}^nZ_iZ_i^T
=\begin{pmatrix}
n^{-1}\sum_iX_i^2&n^{-1}\sum_iX_iY_i\\
n^{-1}\sum_iX_iY_i&n^{-1}\sum_iY_i^2
\end{pmatrix}.}
$$
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