I like to think it in this way: the determinant is the product of the eigenvalues of a matrix, which you can conveniently compute without reducing the matrix to diagonal form. All interesting properties of the determinant are very easy (and often trivial!) to show for the product of the eigenvalues.
More in the spirit of your post, I don’t remember how hard it is to show that the determinant is invariant under unitary transformation, but not too hard I think. It’s not the only invariant of course (the trace is as well, I don’t remember if there are others). But you could definitely start from the product of eigenvalues idea and make it invariant to get the formula for det.
I like to think it in this way: the determinant is the product of the eigenvalues of a matrix, which you can conveniently compute without reducing the matrix to diagonal form. All interesting properties of the determinant are very easy (and often trivial!) to show for the product of the eigenvalues.
More in the spirit of your post, I don’t remember how hard it is to show that the determinant is invariant under unitary transformation, but not too hard I think. It’s not the only invariant of course (the trace is as well, I don’t remember if there are others). But you could definitely start from the product of eigenvalues idea and make it invariant to get the formula for det.
det(AB) = det(A)det(B), so the determinant is invariant to any change of basis, not merely unitary ones:
det(ABA−1) =det(A)det(B)det(A−1) =det(AA−1)det(B) =det(B)