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. CHAPTER 9 9.1 (a) 123026 8101 2850 13xxx (b) 026018 40610[][] 1 0 1 2 0 5 6 2 8850610 10889TAA 9.2 (a) [A] = 32 [B] = 33 [C] = 31 [D] = 24 [E] = 33 [F] = 23 [G] = 13 (b) Square: [B] and [E] Column: [C] Row: [G] (c) a12 = 7 b23 = 7 d32 = does not exist e22 = 2 f12 = 0 g12 = 6 (d) (1) 5815[][] 8 4106010EB (2) [ ] [ ] not possibleAF (3) 321[][] 6 0 420 2BE (4) 28 21 497[ ] 7 14 4914 0 28B (5) 25 13 74[ ] [ ] 36 25 7528 12 52EB (6) 361TC (7) 54 76[][] 41 5328 38BA (8) 92413765TD (9) possiblenot ][][ CA (10) [][] []IBB (11) 66 19 53[ ] [ ] 19 29 4653 46 109TEE (12) [][] 46TCC 9.3 (a) Possible multiplications: 415[][] 8 29929AB 16 4[][] 24 4210AC 71[][]51BC 12[][]2.5 7CB Note: Some students might recognize that we can also compute [B][B] and [C][C]:2 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. 2.5 9[][]1.5 5.5BB 10 6[][]97CC (b) [B][A] and [C][A] are impossible because the inner dimensions do not match: (22) * (32) (c) According to (a), [B][C] [C][B] 9.4 The equations can be rearranged into a format for plotting x2 versus x1: 212130.534 1 66xxxx 048120 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) 2486(7)34 9.5 (a) The equations can be rearranged into a format for plotting x2 versus x1: 212112 0.1121017.4xxxx 0204060801001200 200 400 600 800 If you zoom in, it appears that there is a root at about (404.6, 56.5).3 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. 56.456.4556.556.5556.6404 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 1202(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 1.1(17.4) 10( 2) 0.86D which is relatively small. Note that if the system is normalized first by dividing each equation by the largest coefficient, 12120.11 120.11494 10xxxx the determinant is even smaller 0.11(1) 1( 0.11494) 0.00494D (d) Using Eqs. (9.10) and (9.11) yields 117.4(120) 10(174)404.65120.86x 21.1(174) ( 2)(120)56.511630.86x 9.6 (a) The determinant can be computed as: 1111(0) 1(1) 110A 2212(0) 1(3) 330A 3212(1) 1(3) 131A 0( 1) 2( 3) 5( 1) 1D 4 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. (b) Cramer’s rule 192591110 1 0661xD 20952913100881xD 302 921 93110551xD (c) The results can be checked by substituting them back into the original equations: 2( 8) 5(5) 92(6) ( 8) 5 93(6) ( 8) 10 9.7 (a) The equations can be rearranged into a format for plotting x2 versus x1: 21219.5 0.5 9.4 0.51xxxx 14.414.4514.514.5514.69.8 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 0.5( 2) ( 1)1.02 0.02D (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 19.5( 2) ( 1)( 18.8)100.02x5 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. 20.5( 18.8) ( 9.5)1.0214.50.02x (e) Using Eqs. (9.10) and (9.11) yields 19.5( 2) ( 1)( 18.8)100.02x 20.52( 18.8) ( 9.5)1.024.30.02x The ill-conditioned nature of the system is illustrated by the fact that a small change in one of
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