> I think this is because we introduce a complicated way to calculate determinants and then we use determinants to calculate the eigenvalues?
Yes, the determinant should be taught and defined as the volume of the parallelepiped in n-dimensions defined by the columns of the given square matrix. This perspective makes it immediately obvious that the eigenvalues scale the parallelepiped in each of its dimensions (a basis of eigenvectors makes it even simpler). Of course the volume (determinant) must be the product of these scaling factors (eigenvalues)! Since algebra is too convenient for solving problems, this geometric intuition is often an afterthought if it's even taught at all.
As anecdata this was taught in first year university mathematics for math, engineering, physics, chemistry, etc. students in 1981 in all three universities in Perth Western Australia aka "the most isolated city in the world" [1]
It never occurred to me that geometric parallels would not be given in linear algebra courses.
Yes, the determinant should be taught and defined as the volume of the parallelepiped in n-dimensions defined by the columns of the given square matrix. This perspective makes it immediately obvious that the eigenvalues scale the parallelepiped in each of its dimensions (a basis of eigenvectors makes it even simpler). Of course the volume (determinant) must be the product of these scaling factors (eigenvalues)! Since algebra is too convenient for solving problems, this geometric intuition is often an afterthought if it's even taught at all.