1 CHAPTER 9 9 1 a 0 2 6 x1 8 1 0 1 x 2 2 8 5 0 x3 13 0 2 6 0 1 8 40 6 10 b A A T 1 0 1 2 0 5 6 2 8 8 5 0 6 1 0 10 8 89 9 2 a A 3 2 E 3 3 B 3 3 F 2 3 C 3 1 G 1 3 D 2 4 b Square B and E Column C Row G c a12 7 e22 2 b23 7 f12 0 d32 does not exist g12 6 d 5 8 15 1 E B 8 4 10 6 0 10 28 21 49 4 7 B 7 14 49 14 0 28 2 A F not possible 25 13 5 E B 36 25 28 12 9 2 4 1 T 8 D 3 7 6 5 66 19 T 11 E E 19 29 53 46 54 76 7 B A 41 53 28 38 10 I B B 74 75 52 3 2 1 3 B E 6 0 4 2 0 2 6 C 3 6 1 T 9 A C not possible 53 46 109 12 C T C 46 9 3 a Possible multiplications 4 15 A B 8 29 9 29 16 4 A C 24 4 2 10 7 1 B C 5 1 2 1 C B 2 5 7 Note Some students might recognize that we can also compute B B and C C PROPRIETARY MATERIAL The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation If you are a student using this Manual you are using it without permission 2 10 6 C C 9 7 2 5 9 B B 1 5 5 5 b B A and C A are impossible because the inner dimensions do not match 2 2 3 2 c According to a B C C B 9 4 The equations can be rearranged into a format for plotting x2 versus x1 x2 3 0 5 x1 34 1 x1 6 6 x2 12 8 4 0 0 5 10 15 Therefore the solution is x1 8 x2 7 The results can be checked by substituting them back into the original equations 4 8 8 7 24 8 6 7 34 9 5 a The equations can be rearranged into a format for plotting x2 versus x1 x2 12 0 11x1 x2 10 2 x1 17 4 120 100 80 60 40 20 0 0 200 400 600 800 If you zoom in it appears that there is a root at about 404 6 56 5 PROPRIETARY MATERIAL The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation If you are a student using this Manual you are using it without permission 3 56 6 56 55 56 5 56 45 56 4 404 404 5 405 405 5 The results can be checked by substituting them back into the original equations 1 1 404 6 10 56 5 119 94 120 2 404 6 17 4 56 5 173 9 174 b The plot suggests that the system may be ill conditioned because the slopes are so similar c The determinant can be computed as D 1 1 17 4 10 2 0 86 which is relatively small Note that if the system is normalized first by dividing each equation by the largest coefficient 0 11x1 x2 12 0 11494 x1 x2 10 the determinant is even smaller D 0 11 1 1 0 11494 0 00494 d Using Eqs 9 10 and 9 11 yields 17 4 120 10 174 404 6512 0 86 1 1 174 2 120 x2 56 51163 0 86 x1 9 6 a The determinant can be computed as A1 1 1 1 0 1 1 1 1 0 A2 2 1 2 0 1 3 3 3 0 A3 2 1 3 1 2 1 1 3 1 D 0 1 2 3 5 1 1 PROPRIETARY MATERIAL The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation If you are a student using this Manual you are using it without permission 4 b Cramer s rule x1 x2 x3 9 2 5 9 1 1 10 1 0 D 0 9 6 6 1 8 8 1 5 5 1 5 2 9 1 3 10 0 D 0 2 9 2 1 9 3 1 10 D c The results can be checked by substituting them back into the original equations 2 8 5 5 9 2 6 8 5 9 3 6 8 10 9 7 a The equations can be rearranged into a format for plotting x2 versus x1 x2 9 5 0 5 x1 x2 9 4 0 51x1 14 6 14 55 14 5 14 45 14 4 9 8 9 9 10 10 1 10 2 The solution is x1 10 x2 14 5 Notice that the lines have very similar slopes b The determinant can be computed as D 0 5 2 1 1 02 0 02 c The plot and the low value of the determinant both suggest that the system is ill conditioned d Using Eqs 9 10 and 9 11 yields x1 9 5 2 1 18 8 10 0 02 PROPRIETARY MATERIAL The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation If you are a student using this Manual you are using it without permission 5 x2 0 5 18 8 9 5 1 02 14 5 0 02 e Using Eqs 9 10 and 9 11 yields 9 5 2 1 18 8 10 0 02 0 52 18 8 9 5 1 02 x2 4 3 0 02 x1 The ill conditioned nature of the system is illustrated by the fact that a small change in one of the coefficients results in a huge change in the results 9 8 a The system is first expressed as an augmented matrix 10 2 1 27 3 6 2 61 5 1 1 5 21 5 Forward elimination a21 is eliminated by multiplying row 1 by 3 10 …
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