Assume that $\mu_1, \mu_2$ are vectors ($1\times n$) and $\Sigma_1, \Sigma_2$ are symmetric square matrixes ($n\times n$).
Having $\Sigma$, I want to compute $\mu$ such that : $$ \mu^T \Sigma^{-1} = \mu_1^T \Sigma_1^{-1}+ \mu_2^T \Sigma_2^{-1} $$
How can I do that?
You can't compute $\Sigma$, from this and you should have some other equations for calculation of $\Sigma$, but given $\Sigma$, you can easily calculate $\mu$:
$$\mu^T \Sigma^{-1} = \mu_1^T \Sigma_1^{-1}+ \mu_2^T \Sigma_2^{-1}$$
$$\Rightarrow \mu^T = (\mu_1^T \Sigma_1^{-1}+ \mu_2^T \Sigma_2^{-1})\Sigma$$ $$\Rightarrow \mu = \left((\mu_1^T \Sigma_1^{-1}+ \mu_2^T \Sigma_2^{-1})\Sigma\right)^T$$
$$\Rightarrow \mu = \Sigma^T(\mu_1^T \Sigma_1^{-1}+ \mu_2^T \Sigma_2^{-1})^T$$
$$\Rightarrow \mu = \Sigma^T( \Sigma_1^{-T}\mu_1+ \Sigma_2^{-T}\mu_2)$$
Since, both $\Sigma_1$ and $\Sigma_2$ are symmetric, their inverses are also symmetric:
$$\Rightarrow \mu = \Sigma^T( \Sigma_1^{-1}\mu_1+ \Sigma_2^{-1}\mu_2)$$
Now, if you know that $\Sigma$ is also symmetric, you will have:
$$\mu = \Sigma( \Sigma_1^{-1}\mu_1+ \Sigma_2^{-1}\mu_2)$$