Sample Midterm 3, Math 1554, Fall 2017 Instructors: Dr. Michael Lacey and Dr. Greg Mayer Date: Thursday Oct 26, 8:00 am to 8:50 am PLEASE DO NOT MAKE COPIES OF THIS EXAM @gatech.edu legal name of student (please print) Georgia Tech email (please print) TA's Name (please print) Section Name Instructions for Students If there are questions during the test, students can call/text Dr. Greg Mayer, at 404-621-6464, under supervision of their proctor/facilitator. Students are required to show their work and justify their answers for all questions except where explicitly stated. Organize your work in a reasonably neat and coherent way. Calculators, notes, cell phones, books are not allowed. Please ensure you use a dark pen and clear handwriting. Your exam will be scanned into a digital system. Exam pages are double sided. Be sure to complete both sides. Leave a 1 inch border around the edges of exams. Georgia Tech Honor Code Having read the Georgia Institute of Technology Academic Honor Code, I understand and accept my responsibility as a member of the Georgia Tech community to uphold the Honor Code at all times. In addition, I understand my options for reporting honor violations as detailed in the code. signature date Solutions to this sample midterm will be posted shortly after the last lecture before the midterm. This is intentional: students are encouraged to work on these problems themselves before that lecture. 1. (6 points) Circle true if the statement is true, otherwise, circle false. You do not need to explain your reasoning. (a) If A R22 has complex eigenvalues 1 and 2 , then |1 | = |2 |. true false (b) Matrices with the same eigenvalues are similar matrices. true false (c) If A is a diagonalizable n n matrix, then rank(A) = n. true false (d) The steady state of a stochastic matrix is unique. true false (e) If A is invertible and diagonalizable, then A1 is diagonalizable. true false (f) Any matrix that is similar to the identity matrix must be equal to the identity matrix. true false 2. (1 point) Suppose A is a real 4 4 matrix with eigenvalues 3, 4, and 6i. What is the fourth eigenvalue? 3. (3 points) An n n matrix A satisfies A2 + A = 6In . What are the possible eigenvalues of A? 4. (10 points) If possible, give an example of a matrix that has the following properties. If it is not possible to do so, write 'not possible'. \u0012 \u0013 2 0 (a) A matrix, B, such that B is similar to C = , and B 6= C. 0 1 (b) A 2 2 singular matrix that has an eigenvalue equal to 3i. (c) A 2 2 non-zero matrix, A, that is diagonalizable, but A~x = ~b does not have a solution for all ~b R2 . (d) A 2 2 matrix whose eigenvalues \u0012 are \u00131 = 2\u0012and \u0013 2 = 0, and whose 1 1 corresponding eigenvectors are ~v1 = , ~v2 = . 0 1 5. (5 points) Construct a basis for the eigenspace of A associated with the eigenvalue = 3. 5 1 2 A = 2 2 2 2 1 5 6. (5 points) Consider the Markov chain below. 0.8 0.2 A B 0.8 0.2 (a) Identify the transition matrix, P , for this Markov Chain. (b) Calculate the steady state for the Markov chain. \u0012 7. (10 points) Consider the 2 2 matrix A = \u0013 1 5 . 2 3 (a) Calculate the eigenvalues of A. (b) If possible, construct matrices P and C such that A = P CP 1