Prove that the determinant of a product is the product of the determinants |TS| = |T| |S| in this way. Fix the n × n matrix S and consider the function d: Mn×n → R given by T → |TS|/|S|.
(a) Check that d satisfies condition (1) in the definition of a determinant function.
(b) Check condition (2).
(c) Check condition (3).
(d) Check condition (4).
(e) Conclude the determinant of a product is the product of the determinants.