Experiments with Two-by-Two Matrices

• Show that trace of AB = trace of BA

• Show that (AB - BA)(AB - BA) is scalar

• Find all matrices commuting with .

• Show that A^-1BA can never be diagonal.

• Show that there is no real invertible matrix S with S^-1S triangular.

Spreadsheet in Maple

• Use the spreadsheet in Maple to experiment with the 3x + 1 problem.

• How is the number pattern related to the sucessive derivatives of tan(x) formed?

• How is the number pattern related to the sucessive derivatives of cot(x) formed?

• Express tan(nx) as a rational function of tan(x) for n = 1,2,3,4,5,6.

• Study the number pattern associated with the indefinite integral of the function xⁿe^x.

• List the Chebyshev polynomials T_n(x) for n = 0,1,2,3,4,5,6.

Factoring

• Find an exponent n such that some coefficients appearing in the factorization of xⁿ - 1 are different from 0, -1 and 1.

• Factor (((x² - a)² - a)² - a)² - a - x.