I agree, the way I still see determinant is as the 'volume scaling factor' of a linear transformation.
This means it makes sense that det(A) = 0 means A is non-invertible. It also makes a lot of sense when the jacobian pops up in the multi-dimensional chain rule.
Given the above, and the Cayley–Hamilton theorem, I never really had to know why the determinant was calculated the way it is. The above give enough of an interface to work with it.
Comments
I agree, the way I still see determinant is as the 'volume scaling factor' of a linear transformation.
This means it makes sense that det(A) = 0 means A is non-invertible. It also makes a lot of sense when the jacobian pops up in the multi-dimensional chain rule.
Given the above, and the Cayley–Hamilton theorem, I never really had to know why the determinant was calculated the way it is. The above give enough of an interface to work with it.