Theory: matrices A and B are equivalent if and only if r(A)=r(B).
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Transcript of Theory: matrices A and B are equivalent if and only if r(A)=r(B).
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Obviously, if the ranks of two matrices are equal, they have the same normal forms, and so they are equivalent; conversely, if two matrices are equivalent, their ranks are equal too. That is :Theory: matrices A and B are equivalent if and only if r(A)=r(B).
!!! Please remember:we need to figure out if the ranks of matrices are equal only and we will know if they are equivalent.
Non-degenerate MatrixDefinition: if the rank of square matrix A is equal to its order, we call A a non-degenerate matrix. Otherwise, degenerate matrix.( non-degeneratenon-singular;
degeneratesingular )E----non-degenerate matrix O----degenerate matrix
Theory: A is a non-degenerate matrix, then the normal form of A is an identity matrix E with the same size
EA
The rank of matrix is an important numerical character of matrix.
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Corollary 1: the following propositions are equivalent:
;degenerate-non is)( Ai ;)( EAii singular;-non is )( Aiii
)matrix. elemantaryan is (;)( 21 im PPPPAiv
)()()(Theory
iiiiii )singular.-non is isThat
,0,)((
A
AnAr
:)()( ivii ,EAsuch that ,,,,,, 121 mll PPPPP ,
mll PEPPPPA 121 mll PPPPP 121 :)()( iiiv mPPPA 21
EPPP m21 EA
matrix elementary
)()()()( iviiiiii
Prove:
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Corollary 2:Matrices A and B are equivalent if and onlyif there are m-order and n-order non-Degenerate matrices P,Q, such that nnmmnm QBPA And we also have :If P,Q are non-degenerate, then
r(A) = r(PA) = r(PAQ) = r(AQ)
e.g.).( then ,
301
020
201
,2)(Let 34 ABrBAr
,3)( Br ,degenerate-non is B 2)()( ArABr
The Inverse of a Matrix
.
.1such that ,,0 111 aaaaaa
EBAABBA such that,matrix?,matrix
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Definition : if A is an n-order square matrix, and there is anothern-order square matrix B such that AB=BA=E, we say that B is an inverse of A, and A is invertible.
( 1 ) The inverse of matrix is unique.Let B,C are all inverses of A, then B=EB=(CA)B=C(AB)=CE=C
1ADenote the inverse of A as
( 2 ) Not all square matrices are invertible.
For example
00
01A is not invertible. ,1
dc
baA
0000
01 ba
dc
ba
10
0110
It’s impossible. So A is not invertible.
The questions to answer:
1. When the matrix is invertible?
2. How to find the inverse?
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Review : adjoint matrix nnijaA
nnnn
n
n
AAA
AAA
AAA
A
21
22212
12111ij
ij
a
A
ofcominor
algebraic theis
Adjoint matrix
to?attention paid be should
what , use When we A The order of algebraic cominor!
The adjoint matrix of 2-order matrix A .
dc
baA
ac
bdA
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AA
nnnn
n
n
nnnn
n
n
AAA
AAA
AAA
aaa
aaa
aaa
21
22212
12111
21
22221
11211
A
A
EA AA
EAAAAA
It’s an important formula.
Formula :
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Theory: An n-order square matrix A is invertible if and only if A
AA
11.0A
Prove:,invertible isA For 1 EAA ”“
sideeach oft determinan thefindcan We
111 EAAAA 0 A
,0For A”“ EAAAAA
EAAA
AA
A )1
()1
(
AA
A11 Keep in mind!
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.degenerate-non is
singular-non is invertible is
A
AA
e.g.1..of inverse theFind
dc
baA
AA
A11
ac
bd
bcad
1Solution :
)0( bcad
e.g.2.);,(),(1 jiEjiE ));
1(())((1
kiEkiE
))(,())(,(1 kjiEkjiE
Prove: EjiEjiE ),(),(
),(),(1 jiEjiE By the same method,we can prove others
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That is, the inverses of elementary matrices are elementary matrices of the same size.
——This is the 3rd property of elementary matrices 。Exercises: Find the inverse.
12
11.1 A
21
11.2 B
10
22.3C
12
11
3
1.1 1A
11
12.2 1B
20
21
2
1.3 1C
102
123
111
A
?? ? How to find the inverse of
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Properties of the Inverse
;1
invertible is )( 1
AAAi
;)(,invertible isinvertible is )( 111 AAAAii
;)()( 1 ABEBAorEABiii
;)())(( 11 TT AAiv
;))(( 111 ABABv
).invertible is ,0(,1
))(( 11 AkAk
kAvi
))(()( 11 ABABv E ;)( 111 ABAB
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Methods to Find the InverseMethod 1 : inverse. thefind tomatrix adjoint theUse A
Method 2 :Use elementary operations to find the inverse.
,invertible is invertible is 1 AA sPPPA 211
EAPPP s 211
21AEPPP s
)()( 1operations row
AEEA
.
102
123
111
of inverse theFind 1e.g.
A:
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124100
013210
001111
124100
235010
112001
100102
010123
001111
)(
EA
102320
013210
001111
1A
124
235
112
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?,
153
132
5431
AA
131
7185
112981A
1
001
0001
00001
321
2
aaaa
aa
a
A
nnn
?1 A
Ex
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1000
0010
0001
00001
1
a
a
a
A
Method 3: use the definition.
. Find .0,2 e.g. 11
1
Aaa
a
a
A n
n
:
Guest :
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na
aB
1
1
1
. that prove to
needonly Weright that Is
EAB ?
1AB
:Solution
na
a
1
na
a
1
1
1
E
na
aA
1
1
11
1
1
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. find and
,invertible is that prove2 satisfies let 3..1
2
A
AOEAAAge ,:
EAA 22 EEAA 2)(
EEA
A
2 2
1 EAA
Method 4: prove B is the inverse of A by definition.
: thatprove),integer. positive a is (,Let .4e.g. kOAk 121)( kAAAEAE
))(( 12 kAAAEAE )( 1212 kkk AAAAAAAE
kAE E
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Applications of the inverse—— to solve matrix equations.
.invertible is ,.1 ABAX BAXI 1:Solution
XBPPP s 21EAPPP s 21
)()(
operations RowXEBA
.invertible is ,.2 ABXA
)(
operations elementary of method the:Solution II
sPPPA 211
1:Solution BAXI
)(
.operations elementary
of method the:Solution II
sPPPA 211 XPPBP s 21 EPPAP s 21
X
E
B
A
operationsColumn
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.invertible are ,,.3 CABAXC 11:Solution BCAXI
:Solution II
BAXC 1
1BAAX
When we solve matrix equations, remember that before figuring out the solutions, reduce the matrices at first.
. determine , and , have weIf .1 BBAABA
ABAB ABEA )( AEAB 1)(
, find tooperations elementary use alsocan We
B )()(operations row
BEAEA
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. determine , and , have weIf.2 2 XXAEAXA
))((2 EAEAEAXAX ))(()( EAEAXEA
. then ,invertible is ifOnly EAXEA
).9()3( determine , have weIf .3 21 EAEAA
)9()3( 21 EAEA )3)(3()3( 1 EAEAEA
EA 3