Question

10') Determine the range of a so that the quadratic form Q(x, y, z) = a(r+y+z)+2xy-2yz+2zx positive definite.

Answer 1

vo) Q(X, Y, 2) = q (a+y+2>) +any - 2y2 +22x Q = XTAX where X- & to be positive definite if mateix & minors should be positive A is symmentric A2 let A = b a3 ai is symmentric az c Az Az ai QE C y z b A3 ai z C а, = [x y z axt914+ a₂2 aix+by+ azz 92x + asyt cz -axt by??c2+4*Y [, +9.3+42 [a;+93 + x2[a,+2,] comparing of C = x b=4 asa 292= 2 293 = -2 a = -1 29, = 2 9 = 2 az-| As -1 x 30 condition २ x²-430 & e X a²4 a> ta condition ②

X so => a[~~1] -2[2+1] + [-12-2]50 X 다. - 1 x²x-4-2-2-x>0 x²-24-830 (x-4) (x+2) =0 <>4072 - condemn 3 from conditions , , ③ a>0, x>12, X34, &>-2 if a3-2, a can't be positive according to condition ® so discand -2. x>0 is satiesfied 4 x=4 is not satisfied it satisfies a>0, 4>2 & 43.2. range of ais if x>2, so a 24 is solution 4 cacol for Q(XY, 2) to be positive definiti