Question

1. Prove or disprove the following statements: a. For any matriz A e Rmxn with Rank(A) = r, A and AT have the same fet of singular values. b. For any matrix A € Rnxn, the set of singular values is the set of eigenvalues. 2. Determine the definiteness of the quadratic form (11,12)= 9zł - 40102 + 6z. 3. A hospital trauma unit has determined that 25/6 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 recov covered, 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 tommas O

Answer 1