09/30/04

Theorem Six:

 ~If A is an invertible matrix, then A-1 is invertible and (A-1)^-1 = A

~If A and B are n x n invertible matrix, then so is AB, and the inverse of AB is the product of the inverses A and B in the reverse order, (AB) ^-1 = B-1 A-1

~If A is an invertible matrix, then so is A^ t, and the inverse of A^ t is the transpose of   A-1 so that (A^ t) -1 = (A-1) ^ t

Proof :

Find a matrix c so that A-1C = I and CA-1 = I 

(AB)(B-1A-1) = A(BB-1)A-1 = AIA-1 = AA-1 = I

Thus the product of n x n matrix is invertible, and the inverse is the product of their inverse in the reverse order.

~An elementary matrix- is one that is obtained by performing a single elementary row operation on an identity matrix.

Ex:      0001                                         Hence to make this identity you row switch 1 and 4

            0100

            0010                                        so it is an elementary matrix

            1000

Since row operations are reversible, elementary matrices are invertible

Each elementary matrix E is invertible. The inverse of E is the elementary matrix of the same type that transforms E back into I.