Refer to the definition of elementary matrices in the Mechanics of Matrix Multiplication subsection.
(a) What is the determinant of each kind of elementary matrix?
(b) Prove that if E is any elementary matrix then |ES| = |E||S| for any appropriately sized S.
(c) (This question doesn't involve determinants.) Prove that if T is singular then a product TS is also singular.
(d) Show that |TS| = |T||S|.
(e) Show that if T is nonsingular then |T-1| = |T|-1.