作業二

2-1	Show that trace of AB = trace of BA .
2-2	Show that (AB - BA)(AB - BA) is scalar .
2-3	Find all matrices commuting with .
2-4	Show that A-1BA can never be diagonal.
2-5	Show that there is no real invertible matrix S with S-1S triangular.
2-6	Use the spreadsheet in Maple to experiment with the 3x + 1 problem.
2-7	How is the number pattern related to the sucessive derivatives of tan(x) formed?
2-8	How is the number pattern related to the sucessive derivatives of cot(x) formed?
2-9	Express tan(nx) as a rational function of tan(x) for n = 1,2,3,4,5,6.
2-10	Study the number pattern associated with the indefinite integral of the function xnex.
2-11	List the Chebyshev polynomials Tn(x) for n = 0,1,2,3,4,5,6.
2-12	Find an exponent n such that some coefficients appearing in the factorization of xn - 1 are different from 0, -1 and 1. Reference: 吳振奎，數學中的美--數學美學初探，天津教育，(1996), p. 143.
2-13	Factor (((x2 - a)2 - a)2 - a)2 - a - x. Reference: Ramanujan's Notebooks IV, p. 42.