Yes, but that doesn't prove that the determinant is actually the real signed volume multiplier.
I think this is what led me to feel unhappy about determinants when I was a first-year university student. You need to actually prove that the determinant is the volume of the N-dimensional parallelepiped, and the axiomatic proof doesn't do that.
So you need basically two extra lines after proving those things so that people can say "okay, the determinant eats ignores all input vector non-orthogonality so that it gives volume".
I think this is what led me to feel unhappy about determinants when I was a first-year university student. You need to actually prove that the determinant is the volume of the N-dimensional parallelepiped, and the axiomatic proof doesn't do that.
So you need basically two extra lines after proving those things so that people can say "okay, the determinant eats ignores all input vector non-orthogonality so that it gives volume".