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](http://info.worldbank.org/governance/wgi/index.aspx#doc) 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](http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1148386 "Governance Matters VII: Aggregate and Individual Governance Indicators, 1996-2007") (page 97-99).



Find inverse and determinant of a symmetric matrix - for a maximum-likelihood estimation
Your matrix is diagonal plus rank 1. Use the [Sherman-Morrison formula](https://en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula) and the [matrix determinant lemma](https://en.wikipedia.org/wiki/Matrix_determinant_lemma).


