8.1 This exercise gives an alternative proof of Theorem 8.8 in the 3-dimensional case. If x q(x, y, z) = (xyz) Ay where A = (ajj) is a symmetric 3 x 3 matrix with A1 # 0, 2 # 0, show that Q11 a12 Q31 a32 3 a12 Q13 Z 12 + q(x, y, z) = A1 x + y + Z 2 + x + a11 a11 As Conclude that q is positive-definite if 1, 2, As are all positive, while q is negative-definite if A1 0, 43