1. show that if 1-ab is invertible in a ring with 1 then so is 1-ba

5. (a) Define what it means for a linear transformation T:R" R" to be invertible. 2 marks + 3. C) LAT:2 = bo aliwas nabornia sud done (CD) - \ -- (EU) - 1 -(- [47] 5. (b) Let T: R3 R3 be a linear transformation such that T , and T Find a matrix A such that T(x) = Ax for every x ER. [4 marks] 5. (c) Let T be the linear transformation in part (b). Determine all values of c such that T invertible. (4 marks

2. Let A be an n x n matrix. Which of the following statements are TRUE? (i) If Ax = b has a solution for every beR", then Ax = 0 has non-trivial solutions. (ii) If A is invertible, then Ax = b has a solution for every beR" (iii) If A is invertible, then AP is also invertible. (A) (ii) only (B) (i) and (ii) only (C) (ii) and (iii) only (D) (i) and (iii) only (E) (i), (ii), and (iii)

If f:N N be a function, where N be the set of natural numbers, given
by:

verify that function is an invertible function or not? If it is invertible the also find
the inverse of the function.