I'm reading a quantitative finance pairs trading paper and I'm having some difficulties understanding the transition from page 8 to page 9. Namely, I don't understand how the authors express the conditional likelihood on page 8 to the multivariate normal with $\mathbf{\mu}$ and $\mathbf{\Sigma}$ defined as they are on page 9. I initially thought I'd try to work out the matrix algebra in order to help reveal something. Specifically, I redefined symbolically $\mathbf{\mu} = \mathbf{A}^{-1}\mathbf{b}$ and $\mathbf{\Sigma}=\mathbf{A}^{-1}$. Then I tried this: $$(\mathbf{x}-\mathbf{\mu})^T \mathbf{\Sigma}^{-1}(\mathbf{x}-\mathbf{\mu}) = (\mathbf{x}-\mathbf{A}^{-1}\mathbf{b})^T\mathbf{A}(\mathbf{x}-\mathbf{A}^{-1}\mathbf{b}) = (\mathbf{x}^T - \mathbf{b}^T\mathbf{A}^{-1})\mathbf{A}(\mathbf{x}-\mathbf{A}^{-1}\mathbf{b}) = (\mathbf{x}^T\mathbf{A}-\mathbf{b}^T)(\mathbf{x}-\mathbf{A}^{-1}\mathbf{b}) = \mathbf{x}^T\mathbf{A}\mathbf{x}-2\mathbf{b}^T\mathbf{x}+\mathbf{b}^T\mathbf{A}^{-1}\mathbf{b}$$
Where the second equality follows due to $\mathbf{A}$ being symmetric. But, this didn't lead to anything helpful. Any thoughts?
Thanks