(Solved): Question 6 13 marks The set GL (3) of all invertible 3 x 3 matrices is a group under matrix multip ...

Question 6 13 marks The set GL (3) of all invertible 3 x 3 matrices is a group under matrix multiplication. This question concerns the following subset of GL(3): a ~-{(:. :)) H = 0 10 : a, b € R* a-b 0 b (a) Show that properties SG1 and SG2 in the Subgroup Test (Theorem B24) hold for H as a subset of GL(3). (b) The set H is a subgroup of GL(3) (because property SG3 also holds). This part of the question concerns the mapping : (H, x) ? (R*, ×) a 0 0 0 10 b' a - b 0 b Prove that is a homomorphism. (ii) Find Ker p. (iii) Let r be an arbitrary element of R*. Write down an element A of H such that (A) = r. [4] [4] [4] [1]