Jacobian of
$$y = \Sigma^{-\frac{1}{2}}(\mathbf{x}-\mathbf{\mu})$$
I have done the following:
$y = \Sigma^{-\frac{1}{2}}(\mathbf{x}-\mathbf{\mu})$ and $\mathbf{x} =\Sigma^{\frac{1}{2}}\mathbf{y}+\mathbf{\mu}$.
Taking the first derivative for $\mathbf{x}$ with respect to y gives $d\mathbf{x} = \Sigma^{\frac{1}{2}}$ and the same for $\mathbf{y}$ gives $d\mathbf{y} = \Sigma^{-\frac{1}{2}}$
Therefore,
$$|J| = \begin{pmatrix}0 & \Sigma^{\frac{1}{2}} \\\ \Sigma^{-\frac{1}{2}} & 0 \end{pmatrix}$$
This equals to $1$ but I should get that $|J| = |\Sigma|^{\frac{1}{2}}$ what am I missing?