MATH 136 Spring 2015 Assignment 9 Topics: Elementary matrices, determinants Due: 3:00 pm, Wednesday, July 15th Last Name: First Name: I.D. Number: Section: Mark (For the marker only): Instructions: Submit your solutions in the same order as that of the questions appearing. Your solutions must have legible handwriting, and must be presented in clear, concise and logical steps that fully reveal what you are doing. If you get help or collaborate with someone, then acknowledge the names of those who helped you. Any outright copying of assignments will be reported as an act1 of academic plagiarism. 1 1. Compute the 1 (a) det 2 0 0 4 (b) det 0 0 following determinants. 1 0 0 5 2 3 1 0 1 1 2 2 1 3 0 2 0 1 (c) det(A3 BC T B 1 ) given that A, B, and C are square matrices such that detA = 2, detB = 1, and detC = 3. 2. Compute the determinant of the following matrix by row reducing to an upper triangular matrix. 1 1 2 2 1 1 2 1 4 3. Express the following matrices and their inverses as a product of elementary matrices. 1 2 (a) A = 1 3 1 0 2 (b) B = 0 1 1 2 1 6 4. Let A and B be invertible n n matrices. Prove that detA = detB if and only if A = CB, where C is an n n matrix such that detC = 1. 5. State whether the following statements are true or false and prove or disprove accordingly. (a) If A and B are n n matrices then det(AB BA) = 0. 2 (b) If A is an m n matrix and B is n m then det(AB) = det(BA). (c) If A is an n n matrix then det(AT A) > 0. (d) If A and B are n n matrices and A is not invertible then AB is not invertible. 3