Case 1:

$\displaystyle \int \exp\left( - (Y - A)^{\mathrm{\textsf{T}}}V^{-1} (Y - A) \right) \; dY$	$\displaystyle =$	$\displaystyle \int \exp\left( - Z^{\mathrm{\textsf{T}}}Z \right) \vert\det(V^{-1/2})\vert^{-1} \; dZ$
	$\displaystyle =$	$\displaystyle \left(\pi\right)^{N/2} \vert\det(V)\vert^{1/2}$	(15)

where $\dim(Y) = N$ and using $Z = V^{-1/2} (Y - A)$ so that $dZ = \vert\det(V^{-1/2})\vert dY$ .