18.06- Exercises on positive definite matrices and minima Solved

Problem 27.1: (6.5 #33. Introduction to Linear Algebra: Strang) When A and 

B are symmetric positive definite, AB might not even be symmetric, but its eigenvalues are still positive. Start from ABx = λx and take dot products with Bx. Then prove λ > 0. 

              

Problem 27.2: Find the quadratic form associated with the matrix .

Is this function f (x, y) always positive, always negative, or sometimes positive and sometimes negative?