A set of n × n matrices G ⊂ Mn×n, is said to form a group if
(1) Whenever A, B ∊ G, so is the product A B ∊ G. and
(2) Whenever A ∊ G. then A is nonsingular, and A-1 ∊ G.
(a) Show that I ∊ G.
(b) Prove that the following sets of n × n matrices form a group:
(i) All nonsingular matrices;
(ii) All nonsingular upper triangular matrices;
(iii) All matrices of determinant 1;
(iv) All orthogonal matrices;
(v) All proper orthogonal matrices;
(vi) All permutation matrices;
(vii) All 2 × 2 matrices with integer entries and determinant 1.
(c) Explain why the set of all nonsingular 2 × 2 matrices with integer entries does not form a group.
(d) Does the set of positive definite matrices form a group?