1.5 - 5 Dr Doom Lin Alg MATH185O Section 1 Page 1 of 4...

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Unformatted text preview: 5 Dr. Doom Lin Alg MATH185O Section 1% Page 1 of 4 Chapter 1 - Linear Equations and Matrices Section 1.5- Elementary Matrices and a Method for Finding 14—1. Definition An n X 71 matrix is called an elementary matrix if it can be obtained from the identity matrix In by performing a single elementary row operation. Examples: E\ , 61,65 Are—E Ewnm‘r mm J L Cs my: 1 0 0 1 0 0 E1: 01 ,E2= 0—30 ,E3= 015 ,F: 02 1 0 1 0 0 0 1 0 0 1 ’I ' T. ~—’-9E _ a I fall 3 [1*(4’) 'L 15 (£90481; 3, 22‘ Theorem 1.5.1. Row Operations by Matrix Multiplication If the elementary matrix E results from performing a certain row opera- tion on [m and if A is an m X 71 matrix, then the product EA is the matrix obtained when the same row operation is performed on A. 1 0 0 4 —3 3 Example:WithE= 0 1 0 ,andAz 1 2 —1 , 0 —2 1 2 5 —2 E 152.312.5411 L\ n: 3 EA= l 7. ~\ 0 \ 0 Lt -?> 3 Z l 7. "\ :EA Dr. Doom 5 Lin Alg MATH185O Section 14 Page 2 of 4 Row Operations on I that produce E: 6) Multiply a row 2' by c 7é 0 @ Interchange rows 2' and j @ Add 0 times row 2' to row j Examples: \ o a] __.e o 7— ° 1?: szl, \o a s El 9 \ ——-P I)— (LIQ 2'1 \ O E; \ 'L 15 —’3 \ Row Operations on E that return us to I: Multiply a row 2' by 1/0 Interchange rows 2' and j Add —c times row 2' to row j I I \ 0 0 x J— TE3 : El E~\ “Hew- El :— ° VZ o 7. a. o o \ I ———3 I " E, E wuew = E 2‘9 2‘; 'L z 7. E) Z O \ -I 0 / ,_ I ( ° .__/) t g were“. = ° ° :5 E3 E3 0 o l Dr. Doom Lin Alg MATH185O Section 1.4 Page 3 of 4 Theorem 1.5.2. Every elementary matrix is invertible, and the inverse is also an elementary matrix. SEE 9mm?“ ammE. C. :5, =€4 ) 53 zgs m 0w. emvw. Theorem 1.5.3. Equivalent Statements If A is an n X n matrix, then the following statements are equivalent, that is, they are all true or all false. a) A is invertible. b) Ax = 0 has only the trivial solution. c) The reduced row echelon form of A is [7,. d) A is expressible as a product of elementary matrices. ( ( ( ( Proof: WE‘ LL mow. (515 We“) , (Qw (a), Cc) =>(oU, (d) aaéz.) C“) ‘7 0°) ASSUME A is iiwaimtz. TLCVQL sou-min» «(51's L Ag 4Q \'$ A WEHWS) SM Kurt Au. Mu mu 0? AE=Q‘ £0 AE|=A3=Q Ame-A31— c=> K‘A¥‘=K‘A%Ie’ 1235552. ‘3’ 29:25.. so we «aim. sown'on ls wields. 0’) =78‘) kfis‘m‘i‘é A! 19 We 0151.1 we miui‘m. scum-m. in «at: YMES o? Sowule K we bet 1% espwhem' swam = O . ~ . I . x‘ o 3‘ 'THB $151015 COEeFIctem' Mu (5 In XL = Xu‘o c) egg) Assume Junedfi. (A3 =1 . 50 W m ceramic» in 'NE wLm'on Agave anemones To An manure! Mix. 5 has ‘33“ . ‘31" 2m 30-!) so «a» 221mm. lives A 750—9 EA? EZEA '33? . . . Ek~-.E2E\A =1 of of °" so (€K"' EIE‘Y" (EV-H" €2.6le = (En-“Ezel‘d: C VWC—Té 0‘ I‘N‘WLT‘BLE ) nmmes M; 'mucumu So 1 A = Ef‘e“_,, 6; -x -‘ "\ v u - so A = E‘ Em WE; mum LS A newer ac cum “mules. (51) =2 («3 Assume A CA» (a: amessao As A War as EL‘TIzv Mme-ices s. A: E‘EL.,_,EY_ “L... 555 A0,: 50mm. Since 01°00ch 08 liverhuc MATM‘a—zs A16 [JVtflv‘EWE , A t‘- _/ Dr. Doom Lin Alg MATH185O Section 1.4 Page 4 of 4 Using Row Operations to Find A‘1. We want to find a sequence of elementary row operations that will reduce A to the identity matrix I (if A is invertible), and at the same time perform the same operations on the identity matrix I in order to obtain A‘l. We can do this efficiently by performing this sequence of row operations only once, but on the matrix [ A | I ] (this is the matrix obtained by adjoining I to the right side of A). At the end of the process we will obtain [ I | A‘1 woo) Example: dc! ° w“ A a i a = — 69 \ x 2 4 3 09* 540‘?“ Sad-“A \ 7. \ \ o 0 [Y vi") w“ .. (p 3v \_A\ L's—X = o l -l 0 \ o L H '5 0 o I. A 1 I x o O a \ —’9 O \ "'\ O l O 25323'294 o i "'7. l \ O a fig; L6 WHERE 6}: a l i l 3 a \ ‘ O o a l Q‘tQVrQ'L o l _‘ o l o o o \ —7, o \ _ | 0 -—3 EZE‘A ELE‘ " a I o i .5 Q l i O \ '. +9. 0 a (2i; o l o —z, 1 l o o \ -7. a I 53515 A E31615 {a Q \ o o '7 -7. -3 4 \~ Q.\ 5 I o ‘ o _1 ‘ ‘ = \ A 1 O 0 l —L O \ Z 3 '— E E E — G , 7 ’ — (bit '5 1. M EEC-45. l So A\= Rq— l “\ a“ a" —L o \ ...
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This note was uploaded on 11/10/2009 for the course MATH 1850 taught by Professor Mihaibeligan during the Spring '09 term at UOIT.

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1.5 - 5 Dr Doom Lin Alg MATH185O Section 1 Page 1 of 4...

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