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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 soumin» «(51's L Ag 4Q \'$ A WEHWS)
SM Kurt Au. Mu mu 0? AE=Q‘ £0 AE=A3=Q AmeA31— 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. scumm. 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" (EVH" €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 Mmeices
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 EEC45. 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.
 Spring '09
 MihaiBeligan
 Linear Algebra, Algebra

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