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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).

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Your matrix is diagonal plus rank 1. Use the Sherman-Morrison formula and the matrix determinant lemma.

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$\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}= \begin{bmatrix} \beta_1^2\theta_1^2 & & & \\ & \beta_2^2\theta_2^2 & & \\ & & \ddots & \\ & & & \beta_k^2\theta_k^2\\ \end{bmatrix} + \begin{bmatrix} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_k \\ \end{bmatrix} \cdot \begin{bmatrix} \beta_1 \: \beta_2 \: \ldots \:\beta_k \end{bmatrix} $

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