MATH 136 Spring 2015 Assignment 8 Topics: Coordinates, Matrix Inverses Due: 3:00 pm, Wednesday, July 8th 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 act of academic plagiarism. 1 1. Find the inverses of the following matrices, if they exist. 1 1 (a) A = 1 5 1 2 1 (b) B = 1 2 5 3 2 3 2 1 0 (c) C = 1 0 1 2 2 2 2. (a) Given the bases B = {1, x, x2 } and C = {1 + x + x2 , 1 x, x2 1} of P2 (R), nd C PB . 1 2 1 1 , and C = , of R2 , nd (b) Given the bases B = 1 4 3 2 C PB . 3. Let A be an n n matrix such that AA = kA where k = 0. Prove that A is invertible if and only if A = kI. 4. State whether the following are true or false and prove or disprove accordingly. (a) If A and B are matrices such that AB is invertible then A and B are invertible. (b) If A, B, C Mnn (R) such that A, C, and ABC are invertible then so is B. (c) If A, B Mnn (R) such that AB = B and B = 0 then A is invertible. 5. Let A be an n n matrix, and let B = {v1 , . . . , vn } be a basis for Rn . Prove that the set {Av1 , . . . , Avn } is a basis for Rn if and only if A is invertible. 2