center of G is defined to be Z(G) = {a E G | ag = ga for all ge G}. (a) Show that if X C G is a generating set for G, then Z(G) = {a E G | ax = xa for all x € X}. (b) Find the centers of Sn, Dn, and GLn. (Hint for GLn: There is a theorem from linear algebra that every invertible matrix can be written as a product of ele- mentary matrices. In other words, GLn is generated by elementary matrices.)

Q. 8.(a) Let {e} be an orthonormal set in a Hilbert space H, then Σ|(x,e₁) | ² ≤||x||² for each vector x in H.

Let B be an n x n matrix over a field F, and suppose that B has minimal polynomial x2 + x + 1. (i) Show that B is invertible. (ii) (iii) (iv) Prove that the minimal polynomial of B-1 is also x? + x + 1. Prove that ifF = C and n is odd, then B is not similar to B-1. Prove that if F = R and n is even, then B is similar to B-1.

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