In the video, it mentions that the determinant of a 3x3 matrix represents the volume of the parallelepiped formed by transforming the unit cube aligned with the basis vectors. It explains that the determinant gives the factor by which volumes are scaled, just as determinants in 2D represent the scaling of areas. The video also mentions that a determinant of 0 indicates that all of space is squished onto something with 0 volume, like a flat plane, line, or single point.
Regarding the division aspect, the video doesn't explicitly mention it. However, the concept of subtracting the product of certain elements from the product of other elements in the determinant formula can be understood as a form of division operation. This subtraction or difference in scaling factors is what contributes to the computation of the determinant and reflects the volume change or scaling of the parallelepiped.
It is true that it doesn't answer all the questions one might have about the problem, but it does answer some of them and is likely the video mentioned earlier, which unfortunately wasn't linked by either of you. So, I linked to it so we'd have it here.
> In the video, it mentions that the determinant of a 3x3 matrix represents the volume of the parallelepiped formed by transforming the unit cube aligned with the basis vectors. It explains that the determinant gives the factor by which volumes are scaled, just as determinants in 2D represent the scaling of areas.
Yes, we all already know that. It is explicitly present as assumed background knowledge in the original question that ivan_ah asked:
>>> How does condensation compute the volume of a parallelepiped from the individual areas of parallelogram?
The question is how the area of the parallelogram defined by the two vectors
[e f]
[h i]
is relevant to the volume of the parallelepiped defined by the three vectors
[a b c]
[d e f]
[g h i]
(as I've phrased it, this question also applies to the textbook way of calculating determinants!)
and the 3b1b video not only doesn't address this, or mention that condensation is a thing you can do, it actively discourages you from calculating determinants by any method.
