Consider the matrix :

O 1 A = 0 0 HOR O k (a) Without doing any calculations, look at A find one of its eigenvectors; explain. (b) Find all values for k such that the eigenvalues for A are exactly 1 = 0 and 12 = 2. Choose one such value of k, and diagonalize A. (For this part, show all of your work, but feel free to check your calculations using computational software.) (c) Still using your chosen k, pick one eigenvector v1 for A with eigenvalue )1 = 0 and another eigenvector v2 for A with eigenvalue 12 = 2, and verify that v1 v2 = 0. (How did I know that this would happen without even knowing any of the choices you made?! Read on...) (*) Bonus: Now let B be an arbitrary matrix such that B = B, and suppose that v1, V2 are eigenvectors of B corresponding to eigenvalues )1 = 0 and 12 = 2. Prove that v1 and v2 are always orthogonal. (Don't think about this problem unless you have time it's spicy!)