I recently read the very interesting discussion on matrix determinants here: What's an intuitive way to think about the determinant?

I was hoping to ask a question related to this topic, and decided to start a new question (I hope this is ok!).

Matrix determinants can be thought of as the coefficient by which an oriented volume changes under the linear mapping of that matrix.

For example, consider $Ax=y$. We can consider the entries of $x$ and $y$ as the vertices of an $n$-d parallelopiped. Then $vol(x)$ is the volume of that parallelopiped, and $vol(y)=|det(A)|vol(x)$.

My question truly has 2 parts. The first is whether my example above is a correct interpretation. And the second is, if $x \in \mathbb{R}^n$, how do I compute $vol(x)$?

This interests me, because if finding the "volume" of a vector is easy, than we can use that to find the determinant, i.e.

$|det(A)|=\frac{vol(y)}{vol(x)}$