In a book on convex optimization it is said that if an ellipsoid is defined as
$$\mathcal{E} =\{v:\|Av-b\|_2\le 1\}$$
with $A\in S^n, A \succ 0$ ($A$ is a positive definite matrix and $b$ is a real vector), $b\in \mathbb{R}^n$, then the volume of the ellipsoid is proportional to $\det A^{-1}$. However, I know that the volume of an ellipsoid is proportional to $\det A$, not its inverse. What am I missing here?