Midterm Exam Solutions for First Year Interest Group Seminar | N 1, Exams of Health sciences

Download Midterm Exam Solutions for First Year Interest Group Seminar | N 1 and more Exams Health sciences in PDF only on Docsity! M346 Second Midterm Exam Solution, November 10, 2005 1. Let A = ( 2 −1 2 −1 ) . a) Find the eigenvalues and eigenvectors of A. The determinant is zero, so 0 must be an eigenvalue. The trace is 1, so the other eigenvalue must be 1. The eigenvectors are (1, 2)T and (1, 1)T . b) Compute A13421. (No, you do NOT need a calculator for this!) A = PDP−1, where P = ( 1 1 1 2 ) and D = ( 1 0 0 0 ) . So A13421 = PD13421P−1. But D13421 = D, so this just equals PDP−1 = A. c) Compute eA. eA = PeDP−1 = ( 1 1 1 2 ) ( e 0 00 1 ) ( 2 −1 −1 1 ) = ( 2e − 1 1 − e 2e − 2 2 − e ) . 2. a) In R3 with the standard inner product, apply the Gram-Schmidt process to convert the basis { (1, 4, 3)T , (2, 3, 4)T , (10, 4, 0)T } into an orthogonal basis. y 1 = x1 = (1, 4, 3) T , y2 = x2 − y1 = (1,−1, 1) T , y3 = x3 − y1 − 2y2 = (7, 2,−5)T . b) Find the coordinates of the vector v = (1,−7, 9)T in the orthogonal basis you constructed in part (a). 〈y1|v〉 = 0, 〈y2|v〉/〈y2|y2〉 = 17/3 and 〈y3|v〉/〈y3|y3〉 = −2/3, so v = (17/3)y2 − (2/3)y3. 3. Consider the system of difference equations x(n + 1) = Ax(n), where A =    −2 0 0 −5 1 2 −3 1 0    and x(0) =    1 4 1   . a) Diagonalize A. Eigenvalue −2, 2 and −1, with eigenvectors (1, 1, 1)T , (0, 2, 1)T and (0, 1,−2)T . Note that A is block triangular, and that the sum of the entries in each row is −2. b) Find x(n) for all n. (You may express your answer as a linear combination of the eigenvectors of A, but the coefficients should be explicit.) Since x(0) = b1 + b2 + b3, x(n) = (−2) nb1 + 2 nb2 + (−1) nb3. 4. Consider the nonlinear system of differential equations: dx1 dt = ln(x1x 2 2 ) 1