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If $\mathbf{M} \sim W_2(\Sigma, 3)$ is a Wishart matrix and $\Sigma =\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$ then what is the distribution of $(3, 1) \mathbf{M}^{-1}(3,1)^T$ ?

Thank you!

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There is a theorem which states that if $\mathbf{M} \sim W_d(\Sigma, m)$, and $m > d$, then $\frac{\mathbf{a}^T\mathbf{\Sigma}^{-1}\mathbf{a}}{\mathbf{a}^T\mathbf{M}^{-1}\mathbf{a}}$ has the $\chi_{m-d+1}^{2}$ distribution, for any fixed $\mathbf{a}$.

From this we have that $\frac{1}{\mathbf{a}^T\mathbf{M}^{-1}\mathbf{a}} \sim \frac{1}{\mathbf{a}^T\mathbf{\Sigma}^{-1}\mathbf{a}}\chi_{m-d+1}^{2} =\Gamma(\frac{m-d+1}{2}, \frac{2}{\mathbf{a}^T\mathbf{\Sigma}^{-1}\mathbf{a}}) $.

And from this we have: $\mathbf{a}^T\mathbf{M}^{-1}\mathbf{a} \sim \Gamma^{-1}(\frac{m-d+1}{2}, \frac{\mathbf{a}^T\mathbf{\Sigma}^{-1}\mathbf{a}}{2}) $ where $\Gamma^{-1}$ denotes the inverse-gamma distribution.

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