Elementary Linear Algebra Solutions

Elementary Linear Algebra Solutions - SOLUTIONS TO PROBLEMS...

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Unformatted text preview: SOLUTIONS TO PROBLEMS ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND First Printing, 1991 CONTENTS PROBLEMS 1.6 ............................................ 1 PROBLEMS 2.4 ............................................ 12 PROBLEMS 2.7 ............................................ 18 PROBLEMS 3.6 ............................................ 32 PROBLEMS 4.1 ............................................ 45 PROBLEMS 5.8 ............................................ 58 PROBLEMS 6.3 ............................................ 69 PROBLEMS 7.3 ............................................ 83 PROBLEMS 8.8 ............................................ 91 i SECTION 1 . 6 2. (i) 2 4 R 1 R 2 2 4 R 1 1 2 R 1 1 2 ; (ii) 1 3 1 2 4 R 1 R 2 1 2 4 1 3 R 1 R 1- 2 R 2 1- 2 1 3 ; (iii) 1 1 1 1 1 1 R 2 R 2- R 1 R 3 R 3- R 1 1 1- 1- 1- 1 R 1 R 1 + R 3 R 3 - R 3 R 2 R 3 1 1 1- 1 R 2 R 2 + R 3 R 3 - R 3 1 1 1 ; (iv) 2- 4 R 3 R 3 + 2 R 1 R 1 1 2 R 1 1 . 3.(a) 1 1 1 2 2 3- 1 8 1- 1- 1- 8 R 2 R 2- 2 R 1 R 3 R 3- R 1 1 1 1 2 1- 3 4- 2- 2- 10 R 1 R 1- R 2 R 3 R 3 + 2 R 2 1 4- 2 1- 3 4- 8- 2 R 3 - 1 8 R 3 1 4 2 1- 3 4 1 1 4 R 1 R 1- 4 R 3 R 2 R 2 + 3 R 3 1- 3 1 19 4 1 1 4 . The augmented matrix has been converted to reduced rowechelon form and we read off the unique solution x =- 3 , y = 19 4 , z = 1 4 . (b) 1 1- 1 2 10 3- 1 7 4 1- 5 3- 15- 6 9 R 2 R 2- 3 R 1 R 3 R 3 + 5 R 1 1 1- 1 2 10- 4 10- 2- 29 8- 20 4 59 R 3 R 3 + 2 R 2 1 1- 1 2 10- 4 10- 2- 29 1 . From the last matrix we see that the original system is inconsistent. 1 (c) 3- 1 7 2- 1 4 1 2 1- 1 1 1 6- 4 10 3 R 1 R 3 1- 1 1 1 2- 1 4 1 2 3- 1 7 6- 4 10 3 R 2 R 2- 2 R 1 R 3 R 3- 3 R 1 R 4 R 4- 6 R 1 1- 1 1 1 1 2- 3 2 2 4- 3 2 4- 3 R 1 R 1 + R 2 R 4 R 4- R 3 R 3 R 3- 2 R 2 1 3- 1 2 1 2- 3 2 . The augmented matrix has been converted to reduced rowechelon form and we read off the complete solution x =- 1 2- 3 z, y =- 3 2- 2 z , with z arbitrary. 4. 2- 1 3 a 3 1- 5 b- 5- 5 21 c R 2 R 2- R 1 2- 1 3 a 1 2- 8 b- a- 5- 5 21 c R 1 R 2 1 2- 8 b- a 2- 1 3 a- 5- 5 21 c R 2 R 2- 2 R 1 R 3 R 3 + 5 R 1 1 2- 8 b- a- 5 19- 2 b + 3 a 5- 19 5 b- 5 a + c R 3 R 3 + R 2 R 2 - 1 5 R 2 1 2- 8 b- a 1- 19 5 2 b- 3 a 5 3 b- 2 a + c R 1 R 1- 2 R 2 1- 2 5 ( b + a ) 5 1- 19 5 2 b- 3 a 5 3 b- 2 a + c ....
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This note was uploaded on 03/16/2010 for the course CSE 2300W taught by Professor Shi during the Fall '08 term at UConn.

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Elementary Linear Algebra Solutions - SOLUTIONS TO PROBLEMS...

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