Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 65 / 2 / a

Because $A$ is a real symmetric positive-definite matrix, its inverse exists and all its eigenvalues are positive. Completing the square givesThe translation $v=u-A^{-1}b$ has unit Jacobian determinant. Choose an orthogonal matrix $O$ with $O^{T}AO=\mathrm{diag}(a_1,\dots ,a_n)$, where $a_i>0$. The change $v=Oz$ also has absolute Jacobian one. The remaining integral is the product of $n$ one-dimensional Gaussian integrals, each equal to $\sqrt{2\pi /a_i}$. Consequently the linear-source multivariate Gaussian integral isThe determinant square root is the positive one, as required by convergence of the original real integral.