Assume we have matrices $A_{n\times n}$ and $\Delta_{n\times n}$ and know that:
$$A^{-1}_{i.}\Delta_{.j}\sim N(0,\,A^{-1}_{i.}\,\Sigma_j\,[A^{-1}_{i.}]^\top)\,,$$
where $\Delta\sim N(0,\,\Sigma_j)$, $i$ shows the rows and $j$ the columns; $_{i\,.}$ means row $i$ (vector with dimension $1\times n$) and $_{.\,j}$ means column $j$ (vector with dimension $n\times 1$). Note that this is the distribution of each pair in the $A^{-1}\Delta$ matrix.
Can we derive the distribution of $(A^{-1}_{i.}\Delta_{.j})^2=A^{-1}_{i.}\Delta\,A^{-1}\Delta_{.j}$? Is it a $\chi^2$? If so, what are the parameters?
