> And that the determinant is the volume of the parallelepiped spanned by the vectors:
It always bugs me when people throw that out because it doesn't underscore why that fact is useful. I saw this in my linear algebrea textbook as a practice problem and never got it until 3Blue1Brown spelled it out.
It is quite likely that the determinant was originally defined as the volume of the parallelepiped spanned by the vectors, when the mathematicians were working out linear systems. Then they threw out the intuition and kept the formula for the textbooks. The point is that because the transform is linear, the volume change of the unit square as it is stretched into a parallelepiped is the same transform as will be experienced by any volume in the space, which is quite possibly how the determinant was first identified.
So the perspective where this is an important fact is that the determinant is the volume change of unit volume -> parallelepiped (or higher dimensional equivalent) when a matrix is viewed as a spatial transform, then that turns out to be a remarkably fundamental measure that characterises a lot of useful properties about the entire matrix.