I'm investigating the derivation of the formula for the volume of an $n$-dimensional ellipsoid. I found this in literature: If $S_n$ is an $n$-sphere and we transform it to an ellipsoid using the positive definite matrix $M$, then the volume of the ellipsoid $E_n$ is
$$ V(E_n) = det (M) V(S_n). $$
Can someone explain to me the background of this? By what means can we just take the determinant "out"?
Intuitively it is true. Using the change of variables theorem,
$$\det (M) \cdot V(D_n) = \int_{D_n} \det (M) \, dx_1\cdots dx_n = \int_{M(D_n)} dM(x_1)\cdots dM(x_n) = V(E_n)$$
Note that the Jacobian of the transformation $M$ is $M$ itself.