Suppose that we have three different variables $x$, $y$ and $z$, where $x$ stands for the state of the world and $\mathbb{X}$ is the state space, such that $x\in\mathbb{X}$. The following information hold
$$s= x + \epsilon\quad\text{where $(x,\epsilon)$ are independently distributed and $x\sim N(\mu_x,\sigma^2_x)$, $\epsilon\sim N(0,\sigma_{\epsilon}^2)$}$$
$$y= x + \eta\quad\text{where $(x,\eta)$ are independently distributed and $x\sim N(\mu_x,\sigma^2_x)$, $\eta\sim N(0,\sigma_{\eta}^2)$}$$
and $z$ is correlated with the state of the world $x$ such that $z\sim N(\mu_z, \sigma_z^2)$ and $\sigma(z,s)=\sigma(z,y) = Cov(z,x) = \sigma(z,x) \ne 0$. Note that, $(z,\epsilon,\eta)$ are independently distributed.
Let $I=\{s,y,z\}$ be an information set that contains all three random variables. Then, I want to calculate the following conditional moments $\mathbb{E}(x|I)$, $\mathbb{V}ar(x|I)$ and $\mathbb{C}ov(x,z|I)$.
My idea is to use the projection theorem and hence
$$\mathbb{E}(x|I) = \mathbb{E}(x) + \left(\sigma(x,s), \sigma(x, y), \sigma(x, z)\right) \Sigma_I^{-1} \begin{pmatrix} s-\mathbb{E}(s) \\ y- \mathbb{E}(y) \\ z - \mathbb{E}(z) \end{pmatrix} $$
such that $\Sigma_I^{-1} = \begin{pmatrix} \mathbb{V}ar(s) & \mathbb{C}ov(s,y) & \mathbb{C}ov(s,z)\\ \mathbb{C}ov(y,s) & \mathbb{V}ar(y) & \mathbb{C}ov(y,z)\\ \mathbb{C}ov(z,s)& \mathbb{C}ov(z,y) & \mathbb{V}ar(z) \end{pmatrix}^{-1}$
Then
$$ \mathbb{V}ar(x|I) = \mathbb{V}ar(x) - \left(\sigma(x,s), \sigma(x, y), \sigma(x, z)\right) \Sigma_{I}^{-1} \begin{pmatrix} \sigma(x,s) \\ \sigma(x, y) \\ \sigma(x, z) \end{pmatrix} $$
and
$$ \mathbb{C}ov(x,z|I) = \Sigma_{(x,z)} - \begin{pmatrix} \Sigma_{(x,\{s,y,z\})} \\ \Sigma_{(z,\{s,y,z\})}\end{pmatrix} \Sigma_{I}^{-1} \begin{pmatrix} \Sigma_{(x,\{s,y,z\})} \\ \Sigma_{(z,\{s,y,z\})}\end{pmatrix}^\intercal $$
where $\Sigma_{(x,z)} = \begin{pmatrix}\mathbb{V}ar(x) & \mathbb{C}ov(x,z)\\ \mathbb{C}ov(z,x) & \mathbb{V}ar(z) \end{pmatrix}$, $\begin{pmatrix} \Sigma_{(x,\{s,y,z\})} \\ \Sigma_{(z,\{s,y,z\})}\end{pmatrix} = \begin{pmatrix} \mathbb{V}ar(x) & \mathbb{C}ov(x,y) & \mathbb{C}ov(x,z)\\ \mathbb{C}ov(z,s) & \mathbb{C}ov(y,z) & \mathbb{V}ar(z)\\ \end{pmatrix}$ and the exponent $^\intercal $ stands for the transpose matrix.
The difficult part is to calculate the inverse matrix of $\Sigma_I^{-1}$. I omit the calculations, but I just want to know if I am at the right direction to find the conditional moments? Could anyone please give me some feedback or tell me if there is an easier way to do the calculations? If the dimension was higher, namely I had four random variables, say $s$, $y$, $z$ and $w$ then the calculations become even more difficult, how could someone respond to such a problem?