# This representation of the precision matrix (Inverse of the covariance matrix) confuses me

I am currently reading the book titled "Generalized Least Squares" by Takaeki Kariya and Hiroshi Kurata. In one section, a General linear regression model of the form $$$$y=X\beta+\varepsilon$$$$ where $$y$$ is an $$n\times 1$$ vector, $$X$$ is an $$n\times k$$ known matrix of full rank and $$\varepsilon$$ is an $$n\times 1$$ random vector with mean $$0$$ and covariance matrix $$\Omega$$, such that

$$$$\mathbb{E}(\varepsilon)=0,\quad Cov(\varepsilon)=\mathbb{E}(\varepsilon\varepsilon')=\Omega\in S(n)$$$$ where $$S(n)$$ denotes the set of $$n\times n$$ positive definite matrices. Moreover, $$\Omega$$ is unknown and is fomrulated as a function of an unknown but estimable parameter $$\theta$$ $$$$\Omega=\Omega(\theta)$$$$ A family of models they consider has the following covariance structure: $$$$\Omega=\sigma^2\Sigma(\theta)$$$$ with $$$$\Sigma(\theta)^{-1}=I_n+\lambda_n(\theta)C\quad \theta\in\Theta\subset \mathbb{R}$$$$ where $$C$$ is an $$n\times n$$ known symmetric matrix, $$\lambda=\lambda_n=\lambda_n(\theta)$$ is a continuous real valued function on $$\Theta$$, and the matrix $$\Sigma(\theta)$$ is positive definite for any $$\theta\in\Theta$$.

I follow everything clearly up until the presentation of the precision matrix. I would appreciate clarifying this for me, by the means of an example.

Example: let $$\lambda_n(\theta) = \theta^3$$ and let $$C = \begin{bmatrix}1 & 1 \\ 1 & 1 \end{bmatrix}$$. Then
$$\Sigma(\theta) = (I_n + \lambda_n(\theta) C)^{-1} = \left(\begin{bmatrix}1 \\ & 1 \end{bmatrix} + \theta^3 \begin{bmatrix}1 & 1 \\ 1 & 1 \end{bmatrix} \right)^{-1} = \begin{bmatrix}\theta^3 + 1 & \theta^3 \\ \theta^3 & \theta^3 + 1\end{bmatrix}^{-1}.$$
The class of covariance matrices would be $$\sigma^2 \Sigma(\theta)$$ for $$\theta \in \Theta$$ where $$\Sigma(\theta)$$ has the above form and is positive definite.
For different choices of the function $$\lambda_n(\cdot)$$ and the fixed matrix $$C$$, you will get different sets of covariance matrices.