08bAss2sol - AS2sol/MATH1111/YKL/08-09 THE UNIVERSITY OF...

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Unformatted text preview: AS2sol/MATH1111/YKL/08-09 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1111: Linear Algebra Assignment 2 Suggested Solution 1. (a) Apply the elementary row operations 1 2 R 1 , R 1 + R 2 , R 1 + R 3 , 1 2 R 2 ,- 6 R 2 + R 3 in order, A is reduced into U = 1- 3- 1 1 . (b) Observe that U is upper triangular. Let E 1 = 1 2 0 0 1 0 0 1 , E 2 = 1 0 0 1 1 0 0 0 1 , E 3 = 1 0 0 0 1 0 1 0 1 , E 4 = 1 1 2 1 , E 5 = 1 1- 6 1 . Then, U = E 5 E 4 ··· E 1 A . Take L = E- 1 1 E- 1 2 ··· E- 1 5 , then L = 2 0 0 0 1 0 0 0 1 1 0 0- 1 1 0 0 1 1 0 0 1 0- 1 0 1 1 0 0 0 2 0 0 0 1 1 0 0 0 1 0 0 6 1 is lower triangular; indeed L = 2 0 0- 1 2 0- 1 6 1 . [If you don’t know that the product of lower triangular matrices is lower triangular , prove it now. Hint: Use induction on the order.] 2. Let A = ( a ij ) n × n and B = ( b ij ) n × n . Then, (a) tr( A + B ) = n X i =1 ( a ii + b ii ) = n X i =1 a ii + n X i =1 b ii = tr( A ) + tr( B ) . (b) Straightforward. Skipped. 1 (c) tr( AB ) = n X i =1 n X r =1 a ir b ri and tr( BA ) = n X i =1 n X r =1 b ir a ri . Renaming the indices i and r , it is clear that they are equal. (d) Write A T = ( b ij ) n × n , then b ij = a ij . Now tr( A T ) = n X i =1 b ii = n X i =1 a ii = tr( A ) ....
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This note was uploaded on 12/06/2010 for the course MATH MATH101 taught by Professor Chan during the Spring '09 term at HKUST.

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08bAss2sol - AS2sol/MATH1111/YKL/08-09 THE UNIVERSITY OF...

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