Question

a. For any matrix A € Rmn with Rank(A) = T, A and AT have the same set of singular values. b. For any matrix A € Rnx", the set of singular values is the set of eigenvalues. 2. Determine the definiteness of the quadratic form (11, 12) = 9x - 4x112 +623. 3. A hospital trauma unit has determined that 25% of its patients are ambulatory and 75% are bedridden at the time of arrival at the hospital. A month after arrival, 65% of the ambulatory patients have recovered, 15% remain ambulatory, and 20% have become bedridden. After the same amount of time, 15% of the bedridden patients have recovered, 25% have become ambulatory, 50% remain bedridden, and 10% have died. Determine the percentages of patients who have recovered, are ambulatory, are bedridden, and have died 1 month after arrival. By using a matrix form for this discrete dynamical system, determine the eventual percentages of patients of each type of patient. (Hint: consider the closed formula for each type of patient based on the eigenvalues of the matrix and compute the limit when > 0), 4. In the accompanying table, is listed the scores of five students in the three exams given in a class 7 Hour EXAM 92 NO Stunt KON JNC Pierre Naren WIN Nidterm Exam fi Final Exam 92 90 68 69 61 18 13 TO 08 50
#2

Answer 1

#2. ~! gui²-4042 +6n? The quadratic form q (^,^2) q (0, 2) = (2, M2) That can be written as A ny (4) Where 9 A= (-3 %) 6 Characteristic equation of a 14-40120 A and hence are x²-15d +50=0 70 (-5) = 5,10 eigen positive A is positive definite marrin and Corresponding quadratic form is positive rabies Thus the hence Car definite: Afternative: q (1,42) = 8n) - 2- 34 2 2 + 4 2 12 + 6n2 years (34. - 2 - 2)² + q (min.) 7,0 58 My & 4 N2 ER - q (nime) =0 34 - 3 12 20 20 » China) > Cool q (nine) is positive definite. hin