Let \(\mathbf{X}\) be an \(n\)-dimensional normal random vector with mean vector \(\boldsymbol{\mu}\) and covariance matrix \(\boldsymbol{\Sigma}\), where the determinant of \(\boldsymbol{\Sigma}\) is non-zero. Show that \(\boldsymbol{X}\) has joint probability density

\[ f_{X}(x)=\frac{1}{\sqrt{(2 \pi)^{n}|\Sigma|}} \mathrm{e}^{-\frac{1}{2}(x-\mu)^{\top} \Sigma^{-1}(x-\mu)}, \quad x \in \mathbb{R}^{n} \]