Homework #3 -- Math 307/101, Fall 2011, Joel Friedman PROBLEMS 1 and 2. Do Problem 10 of the Supplemental Exercises, for Problems 6 and 7. (I.e., do Problems 6 and 7 for the G matrix instead of the H matrix.) Your solutions will probably be a function of alpha. Comment on how this function varies as alpha varies from 1 to 0. PROBLEM 3. For arbitrary reals p,q, write out the binomial expansion for (p+q)^4. Write out a similar expansion for (p-q)^4, and use these two expressions to write out expansions for ( (p+q)^4 + (p-q)^4 )/ 2 and ( (p+q)^4 - (p-q)^4 )/ 2 . Show that the matrix A = [ p , q ; q , p ] has p+q and p-q as eigenvalues. Write out an expansion for A^4, and explain its relation to the above expansions involving ( p +/- q )^4 and how diagonalization can be used to understand this relation. [If you really don't want to deal with 4th powers, it's OK to do this with 3rd powers.] Explain what happens with n-th powers. [Note: you are essentially tying down some of the details of class discussion on powers of this A, but you are not assuming that p + q = 1.]