We consider n i.i.d. observations from the structural equation model. We denote by $X := (X_1,...,X_p)$ the $n \times p$ data matrix with n i.i.d. rows, each of them being $N(0, \Sigma_0)$-distributed, where $\Sigma_0$ is a nonsingular covariance matrix. The relations between the variables in a row can be represented as
$$X=XB + E,$$
where $B:= (\beta_{k,j})$ is a $p \times p$ matrix with $\beta_{k,j} =0$ for all j, and where E as an $n \times p$ matrix of noise vectors $E:=(\epsilon_1,...,\epsilon_p)$, with $\epsilon_j$ independent of $X_k$ whenever $\beta_{k,j} \neq 0$. Furthermore, E has n i.i.d rows which are $N(0, \Sigma_0)$-distributed, with $\Omega_0 := \text{diag} (|\omega_1|^2,...,|\omega_p|^2 ) $ a $p \times p$ diagonal matrix. The model implies that
$$\Sigma = [(I-B)^{-1}]^T \Omega [(I-B)^{-1}]$$ How to get the above implication?
My attempt:
Start with the model equation: $X = XB + E$.
Subtract $XB$ from both sides: $X - XB = E$.
Factor out $X$ on the left side: $X(I - B) = E$.
Solve for $X$: $X = E (I - B)^{-1}$.
Compute the covariance matrix $\Sigma$ of $X$. Since $X$ is $N(0, \Sigma_0)$-distributed, we have $\Sigma = \text{Cov}(X) = E[XX^T]$.
What next?