ECON864 Mathematical Economics Session 2 ECON864 Mathematical Economics Session 2, 2012 Assignment 1 Due end of Week 7 Questions from Fundamental methods of mathematical economics, Page 51, Exercises 2 and 5 Page 81, Exercise 1. Also, show that the steady state vector is an eigenvector for the transpose of the transition matrix. Page 88, Exercises 5 and 6 For the matrices in question 5 find the reduced row echelon form. Page 98, Exercises 4, 5, 6, 7, and 8 Page 107, Exercise 3 Page 112, Exercise 3 Page 120, Exercises 2 and 4 1. Let A 2 Rmn, let~ai = (ai1 ai2 ain) be the i th row of A,~cj = 0BBB@ a1 j a2 j ... amj1CCCAthe j th column of A, let En = f~e1; ~e2; : : : ; ~eng be the standard basis for Rn, and let Em = f~e1; ~e2; : : : ; ~emg be the standard basis for Rm. (a) Suppose detA = 1. What does this tell you about m and n? (b) Determine A~ej and~eiTA. (c) Determine~eiTA~ej . (d) Suppose~x 2 Rn is perpindicular to all the rows of A Determine A~x. (e) Suppose~y 2 Rm is perpindicular to all the columns of A. Show that A~x ~y = 0 for all~x 2 Rn. (f) What relation between the number of rows m and the number of columns n will ensure that the homogeneous linear system A~x =~0 has infinitely many solutions? (g) Suppose that for some matrix B the products AB and BA are both defined. What can you say about the number of rows and columns of B? 2. Let A 2 Rnn be a square matrix. As before let~ai = (ai1 ai2 ain) be the i th row of A,~cj = 0BBB@ a1 j a2 j ... an j1CCCAthe j th column of A, and let En = f~e1; ~e2; : : : ; ~eng be the standard basis for Rn. Suppose A is non-singular. (a) What is the span of the columns of A? (b) What is the reduced row echelon form of A? Printed on August 28, 2012 Page 1 of 2Mathematical Economics Assignment 1 3. he characteristic polynomial p(l) of an nn matrix A is a monic polynomial of degree n in l. By the Fundamental Theorem of Algebra p can be completely factored into the product of n linear terms. That is, p(l) = (l??l1)(l??l2) (l??ln); (1) where the li’s are the possibly...
