MATH 1300 D01/D02 Winter 2016 Assignment 4 SHOW ALL WORK to get full marks. Leave answers as exact answers. For example, leave it as fractions such as 1{7 as opposed to decimals such as 0.142857. Word problems should have sentence answers with units. Fractions should be lowest terms. \u0014 \u001c \u0012 \u001a \u0012 \u001a 2 1 3 6 1 7 1 2 , C \u0016 \u0015 3 0 5 \u001d evaluate the following, 1. Let A \u0016 ,B \u0016 3 4 0 2 4 2 5 1 if possible. If not, explain why. [4] (a) AB 5B. [2] (b) pA 5qB. [2] (c) CB. [5] (d) CB T pAB qT [3] 2. (a) Find non-zero 2 2 matrices A and B such that AB [3] \u0016 O. (b) Show that if B is invertible, and AB \u0016 O, then A \u0016 O. Note: (b) does not depend on (a) and the O is the zero matrix, not the number 0. 3. For the following system of equations 2x 3y 4z x 2y 3z y 3z [1] \u0016 3 \u00162 \u00164 (a) Write the coecient matrix A. [6] (b) Use row operations to nd A1 . [3] (c) Use the inverse to solve the system. \u0014 4. The matrix A three steps \u0014 \u0016 \u001c 0 0 1 \u0015 0 1 0 \u001d can be reduced down to the identity in the following 4 0 12 \u001c \u0014 \u001c \u0014 0 0 1 4 0 12 1 0 \u0015 0 1 0 \u001d\u0015 0 1 0 \u001d\u0015 0 1 A\u0016 4 0 12 0 0 1 0 0 [3] [3] [2] [3] 3 \u001c \u0014 \u001c 1 0 0 \u001d\u0015 0 1 0 \u001d\u0016I 0 1 0 0 1 (a) What are the three elementary row operations used in order. (b) Using part (a), nd elementary matrices E1 , E2 and E3 such that E3 E2 E1 A \u0016 I. (c) Write A1 as a product of elementary matrices. Leave your answer as the product. Do not multiply. (d) Write A as a product of elementary matrices. Leave your answer as the product. Do not multiply. 1 5. Let A \u0016 [4] \u0012 5 6 3 4 \u001a \u0012 , 1 D\u0016 0 0 2 \u001a and P \u0016 \u0012 1 1 2 1 \u001a . (a) Compute P 1 and verify that A \u0016 P DP 1 . [2] (b) What is the mistake in the following argument? \"A \u0016 D since A \u0016 P DP 1 \u0016 P P 1 D \u0016 ID \u0016 D.\" [4] (c) Use part (a) to compute A2016 . 6. Suppose there are 3 internet companies, Net A, Net B and Net C. After every month, anybody can choose to stay with their company, or switch to one of the other two. (For simplicity sake, lets assume that no new customers come in, nor any current customer cancel their internet completely.) Data shows after each month 80% of Net A customers stay, 10% switch to Net B and 10% switch to Net C. For Net B customers, 70% of customers stay, 20% switch to Net A and 10% switch to Net C. For Net C customers, 80% of people stay, 15% switch to Net A and 5% switch to Net B. [2] (a) Write down the transition matrix. (First column for A, second column for B and third column for C) [8] (b) After a very long time, what percentage will have Net A, what percentage will have Net B and what percentage will have Net C. This assignment is out of 60 points. 2