3. Let A be an n x n matrix. Let (x, y) = x"y be the scalar product for R". (a) Show that (Ax, y) = (x, A" y) for all x, y € R". (b) Suppose that A is symmetric, i.e., AT = A. Suppose that x and y are eigenvectors satisfying Ax = \x and Ay = µy with A, µ e R and A # µ. Show that (x, y) = 0, i.e., show that x I y.

Use the ratio test to determnine whether an converges, where a, is given in the following problems. State if the %3D ratio test is inconclusive.

(2) Chapter 8 X 4

dt t²Vt² – 16