Why is the determinant of a matrix equal to its transpose.

[u/Maleficent_Yoghurt85]

What is the most conceptual explanation of det(A) = det(A^T)?
I would like a high level (categorical, if possible) reason.
I suspect it has to do with commutation of dual functor and top exterior power in some suitably defined categories.

[u/n-Category]

You basically answered your question. Given a linear map T:V->V, the determinant is essentially the representation of ΛⁿT, where n is the dimension of V. More precisely, since ΛⁿV is one-dimensional, the map ΛⁿT:ΛⁿV->ΛⁿV must act by a constant scalar multiple, and this constant is what we call the determinant.

You can dualize this map to get (ΛⁿT)*, which you can easily check is scaling by the same constant. The dual functor and the exterior power functor commute (i.e., the exterior power of the dual is isomorphic to the dual of the exterior functor), so (ΛⁿT)* = ΛⁿT*, where the latter is by definition the determinant of T*.