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# Engineering Mathematics MATH 3321

UH

GPA 3.69

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This 49 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 3321 at University of Houston taught by Jeffrey Morgan in Fall. Since its upload, it has received 40 views. For similar materials see /class/208395/math-3321-university-of-houston in Mathmatics at University of Houston.

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Date Created: 09/19/15

Nonsingular Matrices and Determinants cont WWWlll lWIHilllll llll llll lml ll smug Pmm MthJZI I Iumvcr mm Mam mmnmm mu 27m a o o a o quotcocoa Popper 04 1 Give the determinant of 0 1 2 3 4 None of these ecall 1 2 0 g 010613 Example Gwe the detenmnant of 1 0 2 L0 039 1 3 2quot R 2 2 4 1 040 QMQZFUA O 44h 3 24 4 o quotquot39quot 739 o o 2 395 M 7 1 u M o I 0 l 3 J I EVA 33 MW ustii 02 Ht Is there a geometric iilnlterplretatiolnl of the all e t e Ir Im ii H a n t 0 f a n In x H m at r ii 4 ea R we m 1 a W Foam new boo 39m R 043m Volume Ks Vwmy 39ustiin A W How do ellelmelnutalry Irow operations Hem the dltelrlmimalnlt of a matrix justii C nit How is the iiea of deterlm ii Inlant Ireated to iinlealr independemce W 1 Ali mm y z fii n ii39tii n An Inlm matrix A is nnsingula r if alnlcll Ul llly if An nxln matrix A is nnsingulm if and O39lnllly iif tuA 33 N A3 3 13 g o A WadiGibb Q 02 Ax 6b 10 How is the dletlrlmiihaht of a matrix Iretatect to the etelrlmiilnlalnlt at the the itlnlvetse at the matrix iiif iit exists if the tlrahspose I OH Dad AT I use of the matrix if 11 ues n Suppose A and B are II39IIXH matrices and a a scaar eterm ii He whiich of the f00wiilnlgm are Dru detA B i detA detlB dletA B dletA detB deta A detA Mo U 12 Popper 04 a quot 0 2 The determinant of the 3X3 matriXA is 43 i O O 0 What is the determinant ofA 1 O O O D o o a 34 O O O O o o b 34 O O c 43 d None of these Popper 04 3 The determinant of the 5X5 matrix A is 43 What is the determinant of A T a 34 nlrr 0 oo oo oo 039 900 00 00 voo voo b 34 c 43 1 None of these Popper O4 mullrr 4 The determinant of the 5X5 matrix A is 43 and the determinant of the 5x5 matrix B is 32 Compute detB TA IB a 24 000000000 0000010000 b 32 c 83 1 None of these Discovery Activity Act the matrix 2 1 A 1 2 on the unit circle and discuss any observations Next Time Steady States for Autonomous Nonlinear Systems Z 4A Vic0 r 69 WW Stability of Steady States for Autonomous quotquotquotquot Nonlinear Systems 7 T Sinid I g W 9 1 43 ng H i 092 i39a L 3W 0 c gtr 0 WW to gt03 QHQ 7 gar de e r Unghbu T id gowcc i chm x 0er 6394 2235 En w 46 L03 Sodom Z Qource av n quot MgrL Ericka t gine 4 6R 5 a 5M9 9 MD w r 4103 Ks muo 993 CG 59 Sgk Jm cm a 0 ML Ks 1 S30 14on 31L Meekquot me l39aauoH S www 4W V awwc 037 do star all 7 O N W MAAa oU 3 a K80 SW19 Linearized Stability 2 1 n H39flfur 9 3 39R Spg L Ste 0 b 0 1de gme i2 pid 39 a C we 40432 Example The Brusselator 7 p H1 39El f HHJ rm H H i bu ENEH all 3970 HICQLMB UL a z ltA Ul OHM Law WA I my gush SW4 0le 3 a 1 103U OQ U2 3390 Ml 39 1 1 1 MI 1151me ME 39 E M1 39 cm L ME I J C39 U0 7 UL C g ll 2R VH3 I autuz U 3 OU gulf U7 59 351 L W x aux U 1 gt31 99 sm Av Popper 23 1 Give the number of steady states for the system 1139 2 u v 7 v39vu 057 wa I Steady States and Stability Reminder Midterm InClass Portion March 13th TakeHome Portion Due March 25th I Power I rsnm mmml Sunkm IMIIIIIIIIWIIIIIl llll lllmmll i lmlllll 1 7an w nmanu 1 WV ooooooooto ononooououonnuououoooononv coo ooo coo ooo ooo ooo ooo Coo coo ooo ooo ooo ooo ooo coo ooo coo coo ooo coo ooo coo ooo ooo ooo We discussed 0 De nitions f Examples Separable Equations 0 Linear Equations 0 Autonomous Equations and Steady States Pepper 12 2 7quot C a In 1 lee the value of the solut10nyt to y 39 y W Z 1 3 When t 2 m A 3 t a I Z a e 393 8 WW 16 a O E C 26 7 r 62 k r Q e None of the above 3 c 7 e w 0 C 3t 2 8 3 1y 3 61 3 Q6 K o Popper 12 2 Give the value of the solu onyx m y 39 xy y0 6 when x 2 33 114 c5 d6 eN one of the above Autonomous First Order Differential Equations g39 gj Mimi MM 4 on Jana cm 1 WWW L12 5 0M3 Steady Qfofnqo 1 Latvuo I J gwsi39wjk39 0 4 r I 501M 739 397 3 0 9 39 3 4 Stability 3 j 1 a wax Wait mgr g 3 a Shun 8 47 SMQ EMA gtO W I Q 34 OM K 360 5ltg W EJo jlt rwtao glt x 31 S ECQ M swim95 ggvo 89 M ii y ltg m Vi ag 05 JL ibo Popper 12 3 Give the number of steady state solutions toy 39 y if a 3 b 2 c 1 d 0 e None of the above Example Discuss steady states stability and forward time behavior of solutions for the differential equation 5quot Ulrwa hailich 7 680 UAW Q50 9 H dx 9 8f 3 quot19 31 W39 sh wk gwaau gm 091 i30 3 quot 3m 0 3 0 M 21W Jaw Sud it 0 34 M a q K 450 SW A 371 H cplquot 03 JVJ M g 0 l5 an un a Sftaa 3 cl 35 an S aae 5MQ WIS quot L C F QJ M IS gtAo4 k 0quot 0 a s hu e Q53quot 1 8449 u PIt 6 JJM 0 4 0A t o UV 9 unS hue U s hu asaw o S40 cquot 0 kaljgy449 0 vv 94 ear 00030000 i0 I I l C 80 o Popper 12 4 Classify the largest steady state solution toy 39 y if a unstable 1 asymptotically stable 0 stable d it is not possible to classify the steady state e None of the 3 ve Example Discuss steady states stability and forward time behavior of solutions for the differential equation y39yb y2 for various values of the parameter b A Nat W 11 Linearized Stability 12 Example Use linearized stability to discuss the steady states stability and forward time behavior of solutions for the differential equation Example Use linearized stability to discuss steady states stability and forward time behavior of solutions for the differential equation y39yby2 for various values of the parameter b Honors Math 3321 Jeff Morgan j morganmathuhedu http Wwwmathuh edujmorganf1Iath3 32 1 T QOf S Sfeln Recall as 0 39 Vquot 39 h 7 1 eg 0 1 Augmented Matrix U7Lr ax VL A OHan 3 b1 Ll l A a17 ig F A azwxquot b1 o 1 Qm7 lr QM397L239 quot3r OK A zbm 2 Elementary Row Operat10ns m 5 3 4 Kj R 2S Question HOW do a 39 a o eeInentary row operatmns R39k J fF A 0 f 1 7 K4 3 f K g Recall We used elementary row operations to solve 3x10y4z 1 2x 7y4z2 4x 9yllz15 Solutionzx 1y Oz 1 Example Vlb QN WWN MALer Use elementary row operations to solve amp Y N0 1 3x10y4z1 Cdcujphr Y1 2x7y4z2 Note A student on class informed me that we entered a number wrong Work through it to get the correct info Row Reduced Echelon Form RREF 545 1 3quot W 3 I Vlntivaling Example 9 ffw MS quot5 54 myquot ZI I3X 3t I 2 24l W39 10 Jxzx l 0 2 x 3 7 2 v o a quot39In quotIn I z 4 l 4 o I D In Ills ilOSIIf ool lrs 3 Q ULmX39 447quot w quot 41 uS 7 hawk 39 9 7 a m M Ti s a 3 L 33 r T 3 x lt 3 9 3 3 5 Im p h I g 5 1 390 u x 4 1quot 9 quot t B quot w quotn 3 Irvquot WW quot4 umquot u K 0in IW V X Can 39039 quotquot339 I used the Matrix Calculator linked from h ponlinemathuhedu The augmented matrix is 2R2 URI gt R1 gives 2 3 3 1 2 1 0 187 17 17 1 2 4 1 1 0 1 57 37 47 1 0 3 2 1 0 0 397 157 87 R1 ltgt R2 gwes 739R3 gt R3 gives 1 2 4 1 1 1 0 187 17 17 21 33 33 i 0 1 57 37 47 0 0 1 513 839 2R1 1R2 gt R2 gives 57R3 1 R2 gt R2 gives 1 2 4 1 1 1 0 187 17 17 0i 70 3 14 0 1 0 213 2839 0 0 1 513 839 URI 1R3 gt R3 gives 187R3 1R1 gt R1 gives 1 2 4 1 1 1 0 0 1113 513 0 7 5 3 4 0 1 0 213 2839 0 2 7 3 0 0 0 1 513 839 17R2 gt R2 gives 1 2 4 1 1 0 1 57 37 47 0 2 7 3 O 2R2 1R3 gt R3 gives 1 2 4 1 1 O 1 57 37 47 0 0 397 157 87 ROW Reduced Echelon Form RREF General Process Info Imeinarow VIM 291 0 I a 81 W5 o NW 3 Eva HrWMquot WW 39 leadih Lek VA a 0x1 15 9 14m 1 A WPWFM 6 m lh 0 hp6 0 gm Lw ohms L You 3 l Ey mei NOON M MW RadMC 015 0 ZiaO a 3 4mm are 0 myef Mow 3410 MWS Z ma3 f h 19 am 3RZVA 9R WW M14 Mu0 Example The RREF of the augmented matrix for the linear system is Solve the system 1O

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