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Problem session exercises 1. A function f (A, B), where A and B are n × n-matrices, is defined by the formula (a) f (A, B) = tr(AB); (b) f (A, B) = tr(AB T ); (c) f (A, B) = tr(AB) + tr(A) tr(B). For each of these functions, find out which of the following properties hold (for all matrices): (1) f (A, B) = f (B, A); (2) f (A, B1 + B2 ) = f (A, B1 ) + f (A, B2 ), (3) f (A, A) > 0 for A 6= 0. 2. Give an example of a n × n-matrix A for which “there is no square root”, that is the matrix equation B 2 = A has no (complex) solutions. (Hint: look among matrices A with An = 0.) 3. Let G = Sn and let T = {1, 2, . . . , m} where 0 ≤ m ≤ n. (a) Explain why |GT | = (n − m)! and |G(T ) | = m!(n − m)! . (b) Is GT a subgroup of (G(T ) , ◦)? 4. Find the Cayley table for (D3 , ◦), the dihedral group of order 6 (that is, the symmetry group of the equilateral triangle with centre 0 and one vertex at e1 ).