Solution
= Solution
Stationarity makes the covariance depend only on $\tau=t-s$. For $\tau>0$,
$$
G(\tau)=e^{-\tau A}\sigma.
$$
At the reversed lag, the $s>t$ formula gives
$$
G(-\tau)=\sigma e^{-\tau A^T},
$$
and therefore
$$
\boxed{G(\tau)=G(-\tau)^T}.
$$
The same identity follows directly by exchanging the two random variables in $G_{ij}(t,s)=\langle x_i(t)x_j(s)\rangle$.