Solution
= Solution
Conditionally on $X$, the log likelihood differs from
$$
-\frac1{2\sigma^2}(Y-X\beta)^T\Sigma^{-1}(Y-X\beta)
$$
only by terms independent of $\beta$. <Differentiation> gives the weighted normal equations
$$
X^T\Sigma^{-1}X\widehat\beta=X^T\Sigma^{-1}Y,
$$
and therefore
$$
\boxed{\widehat\beta(\Sigma)=(X^T\Sigma^{-1}X)^{-1}X^T\Sigma^{-1}Y.}
$$