Determinant of a Product Complete the proof that the determinant of a product is the product of the determinants, using the results of the previous problem and, from Sec. 3.3, Problems 40 and 41.
(a) Show that if A is not invertible, then |AB| = |A| | B |.
By Problem 34 in Sec. 3.3, if A is not invertible, then neither is AB.
(b) When A is invertible, you should use the fact that
AB = (EpEp-1 · · · E1I)B,
Where each Ej represents an elementary matrix for a row operation, to show that |AB| = |A| |B|. First show that |AB| = (-1)sk1 k2 · · · ks| B | for some integer s and constants k1, k2, . . . , ks?