MATH 240 Final: MATH240 BOYLE-M SPRING2012 0101 FINAL SOL

• School: University of Maryland
• Department: Mathematics
• Course: MATH 240
• Professor: All

Document Summary

Math 240 spring 2012 final exam solutions: for x = (x1, x2) in r2, de ne q(x) = 4(x1)2 + 6x1x2 + 3(x2)2. (a) (15 pts. ) Determine whether this quadratic form is positive de nite, positive semide nite, negative de nite, negative semide nite or inde nite. Q(x) = xtrax for the symmetric matrix a = (cid:18)4 3 mial of a is t2 7t + 3, which has roots (by the quadratic formula) The characteristic polyno- (7 p( 7)2 4(3))/2 = (7 49 12)/2 = (7 . There are two positive roots, and therefore q(x) is positive de nite. The minimum value is the smallest eigenvalue, which is (7 37)/2 . (c) (15 pts. ) The vectors (cid:0)1 2 3(cid:1) ,(cid:0)1 2 4(cid:1) and (cid:0)2 1 1(cid:1) are the corners of a tri- angle t in r3. The area of t equals the area of the triangle t obtained by translating t by.