So, this is not exactly a rigorous proof, but off the top of my head, I would justify/remember the properties like: identity has determinant 1 because it doesn’t change the size of any volumes, determinant is alternating because swapping two axes of a parallelpiped is like reflecting those axes in a mirror, changing the orientation. Multilinearity is equivalent to showing that the volume of a parallelpiped A=a1∧a2∧(...)∧an is linear in all the ais. But this follows since the volume of A is equal to the volume of a2∧(...)∧an multiplied by the component of a1 projected onto the axis orthogonal to a2,...,an, which is clearly linear in a1. This last fact is a bit weird, to justify intuitively I imagine having a straight tower of blocks which you then ‘slant’ by pulling the top in a given direction without changing the volume, this corresponding to the components of a1 not orthogonal to a2,...,an.
So, this is not exactly a rigorous proof, but off the top of my head, I would justify/remember the properties like: identity has determinant 1 because it doesn’t change the size of any volumes, determinant is alternating because swapping two axes of a parallelpiped is like reflecting those axes in a mirror, changing the orientation. Multilinearity is equivalent to showing that the volume of a parallelpiped A=a1∧a2∧(...)∧an is linear in all the ais. But this follows since the volume of A is equal to the volume of a2∧(...)∧an multiplied by the component of a1 projected onto the axis orthogonal to a2,...,an, which is clearly linear in a1. This last fact is a bit weird, to justify intuitively I imagine having a straight tower of blocks which you then ‘slant’ by pulling the top in a given direction without changing the volume, this corresponding to the components of a1 not orthogonal to a2,...,an.
Yeah, that’s what I mean by saying it’s “obvious”. Similar to the change of variables theorem in that way.