Evaluate the determinant $\det \Omega $ and find the inverse matrix $\Omega^{-1}$ of:

$$\Omega = \begin{bmatrix} \beta_1^2(1+\theta_1^2) & \beta_1 \beta_2 & ... & \beta_1 \beta_{k-1} & \beta_1 \beta_k \\ \beta_2 \beta_1 & \beta_2^2(1+ \theta_2^2) & ... & \beta_2 \beta_{k-1} & \beta_2 \beta_k \\ ... & ... & ... & ... & ... \\ \beta_k \beta_1 & \beta_k \beta_2 & ... & \beta_k \beta_{k-1} & \beta_k^2(1+\theta_k^2) \end{bmatrix}$$

**Application**

I got this problem when attempting to understand the methodology of the Worldwide Governance Indicators where the authors specify a log-likelihood function of three unknown parameters to solve the maximisation at page 97-99 here Governance Matters VII: Aggregate and Individual Governance Indicators, 1996-2007 (page 97-99).