If A and A I are invertible matrices a show that AA I 1 I A1

If A and A I are invertible matrices,

(a) show that A(A I)^ 1 (I A^1 ) = I

(b) is I A^1 invertible? If yes, find the inverse; if no, give your reason.

Assume A is invertible. Then A^-1 exists and we have

(A^-1)TAT = (AA^-1)T = IT = I

So AT is invertible and (AT )^-1 = (A^-1)T

b)solution -:

A1(y)=(the unique x in X such that A(x) = y)

A1(T(x)) = x, for all x in X, and

A(A1(y)) = y, for all y in Y .

If a function A is invertible, then so is A1,

(A1)1 = A so inverse will be A it self