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I've seen many sources about the geometric meaning of determinants, such as the 3blue1brown's "Essence of Linear Algebra" playlist and this video. However, I couldn't find anywhere the relationship between the algebraic expression for calculating the value of the determinant of an $ n \times n$ matrix and the signed "hypervolume" of the unit cube of $ \mathbb{R^n} $ after the transformation done by the $n \times n$ matrix.

At most, the sources would only cover cases such as $n=1,2 \text{ or } 3$, which are easy to be dealt with. But what I really would like to know is how the full algebraic expression for calculating the determinant of an $n \times n$ matrix connects to the signed volumes of transformed hypercubes of $\mathbb{R}^n$ for all $n \in \mathbb{Z}^+_*$.

Clarification: by algebraic expression for calculating the determinant I mean the use of something like Laplace's expansion $\det(A_{n \times n}) = \sum\limits_{i=1}^n (-1)^{i+j} M_{ij}$. So what I want is to, given an intuitive definition of something like a "hypervolume" of the "parallelogram" at $\mathbb{R}^n$ whose points include the origin and each of the matrice's columns, arrive at this expression for the determinant.

Arthur Queiroz
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  • I'm not really sure what you're looking for, but perhaps you'll find [this post](https://math.stackexchange.com/q/668/81360) helpful. – Ben Grossmann Mar 19 '22 at 00:17
  • you may find these [few pages](http://www.owlnet.rice.edu/~fjones/chap8.pdf) interesting – G Cab Mar 19 '22 at 00:29
  • You can use the definition of the determinant in terms of a multilinear function of the columns and convince yourself that each column operation is compatible with the concept of a signed area – Chris Sanders Mar 19 '22 at 02:22
  • But here's the thing: the intuition is mostly in the visualisation or the geometry of what a volume is. As soon as you ask for a concrete algebraic definition, it's not going to be totally intuitive – Chris Sanders Mar 19 '22 at 02:24

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