excellent course of liaoning chapter 2 matrix the inverse of a matrix & the adjugate matrix
TRANSCRIPT
![Page 1: Excellent Course of Liaoning Chapter 2 Matrix The Inverse of a Matrix & the Adjugate Matrix](https://reader031.vdocuments.us/reader031/viewer/2022020219/56649e8f5503460f94b93c79/html5/thumbnails/1.jpg)
Excellent Course of Liaoning
Chapter 2 Matrix
The Inverse of a Matrix & the Adjugate Matrix
![Page 2: Excellent Course of Liaoning Chapter 2 Matrix The Inverse of a Matrix & the Adjugate Matrix](https://reader031.vdocuments.us/reader031/viewer/2022020219/56649e8f5503460f94b93c79/html5/thumbnails/2.jpg)
Excellent Course of Liaoning
2.3 The Inverse of a Matrix & The Adjugate Matrix
2.3.1 Inverse MatrixDefinition 2.9
1n nE E Notes:
If A is invertible, 1 1AA A A E
If B is an inverse of A , then A is an inverse of B.
![Page 3: Excellent Course of Liaoning Chapter 2 Matrix The Inverse of a Matrix & the Adjugate Matrix](https://reader031.vdocuments.us/reader031/viewer/2022020219/56649e8f5503460f94b93c79/html5/thumbnails/3.jpg)
Excellent Course of Liaoning
Theorem 2.2
If A is invertible, then its inverse is unique.
Proof
Suppose B and C are both inverses of A. We have
,AB BA E AC CA E
then
B BE B AC BA C EC C
![Page 4: Excellent Course of Liaoning Chapter 2 Matrix The Inverse of a Matrix & the Adjugate Matrix](https://reader031.vdocuments.us/reader031/viewer/2022020219/56649e8f5503460f94b93c79/html5/thumbnails/4.jpg)
Excellent Course of Liaoning
2.3.2 Adjugate Matrix and a Formula to Find InverseDefinition 2.10
!!!
![Page 5: Excellent Course of Liaoning Chapter 2 Matrix The Inverse of a Matrix & the Adjugate Matrix](https://reader031.vdocuments.us/reader031/viewer/2022020219/56649e8f5503460f94b93c79/html5/thumbnails/5.jpg)
Excellent Course of Liaoning
Find the adjuagte (classical adjoint) of AExample 2-5
a bA
c d
A
- cd - b
a
Solution
![Page 6: Excellent Course of Liaoning Chapter 2 Matrix The Inverse of a Matrix & the Adjugate Matrix](https://reader031.vdocuments.us/reader031/viewer/2022020219/56649e8f5503460f94b93c79/html5/thumbnails/6.jpg)
Excellent Course of Liaoning
Theorem 2.3
Proposition 2.1
![Page 7: Excellent Course of Liaoning Chapter 2 Matrix The Inverse of a Matrix & the Adjugate Matrix](https://reader031.vdocuments.us/reader031/viewer/2022020219/56649e8f5503460f94b93c79/html5/thumbnails/7.jpg)
Excellent Course of Liaoning
Proof
![Page 8: Excellent Course of Liaoning Chapter 2 Matrix The Inverse of a Matrix & the Adjugate Matrix](https://reader031.vdocuments.us/reader031/viewer/2022020219/56649e8f5503460f94b93c79/html5/thumbnails/8.jpg)
Excellent Course of Liaoning
Example 2-6
Solution
1 2 3
| | 2 2 1 2 0
3 4 3
A
11
21
31
2
6
4
A
A
A
12
22
32
3
6
5
A
A
A
13
23
33
2
2
2
A
A
A
![Page 9: Excellent Course of Liaoning Chapter 2 Matrix The Inverse of a Matrix & the Adjugate Matrix](https://reader031.vdocuments.us/reader031/viewer/2022020219/56649e8f5503460f94b93c79/html5/thumbnails/9.jpg)
Excellent Course of Liaoning
11 21 31*
12 22 32
13 23 33
2 6 4
3 6 5
2 2 2
A A A
A A A A
A A A
then
1 *
1 3 21
3 2 3 5 2| |
1 1 1
A AA
1 1A A AA E
![Page 10: Excellent Course of Liaoning Chapter 2 Matrix The Inverse of a Matrix & the Adjugate Matrix](https://reader031.vdocuments.us/reader031/viewer/2022020219/56649e8f5503460f94b93c79/html5/thumbnails/10.jpg)
Excellent Course of Liaoning
Example 2-7
Proof
![Page 11: Excellent Course of Liaoning Chapter 2 Matrix The Inverse of a Matrix & the Adjugate Matrix](https://reader031.vdocuments.us/reader031/viewer/2022020219/56649e8f5503460f94b93c79/html5/thumbnails/11.jpg)
Excellent Course of Liaoning
Theorem 2.4
Proof
![Page 12: Excellent Course of Liaoning Chapter 2 Matrix The Inverse of a Matrix & the Adjugate Matrix](https://reader031.vdocuments.us/reader031/viewer/2022020219/56649e8f5503460f94b93c79/html5/thumbnails/12.jpg)
Excellent Course of Liaoning
Example 2-8
Proof
![Page 13: Excellent Course of Liaoning Chapter 2 Matrix The Inverse of a Matrix & the Adjugate Matrix](https://reader031.vdocuments.us/reader031/viewer/2022020219/56649e8f5503460f94b93c79/html5/thumbnails/13.jpg)
Excellent Course of Liaoning
Definition 2.11
![Page 14: Excellent Course of Liaoning Chapter 2 Matrix The Inverse of a Matrix & the Adjugate Matrix](https://reader031.vdocuments.us/reader031/viewer/2022020219/56649e8f5503460f94b93c79/html5/thumbnails/14.jpg)
Excellent Course of Liaoning
2.3.3 Formulas for Inverse Matrices
1 11 ( )A A
1 1 12 ( )AB B A
1 1 1 11 2 2 1( )k kA A A A A A
1 13 ( ) ( )n nA A
1 114 ( )kA A
k
1 15 ( ) ( )T TA A
1 16 | |
| |A
A
![Page 15: Excellent Course of Liaoning Chapter 2 Matrix The Inverse of a Matrix & the Adjugate Matrix](https://reader031.vdocuments.us/reader031/viewer/2022020219/56649e8f5503460f94b93c79/html5/thumbnails/15.jpg)
Excellent Course of Liaoning
2.3.4 Formulas for Adjoint (Adjugate) Matrices
![Page 16: Excellent Course of Liaoning Chapter 2 Matrix The Inverse of a Matrix & the Adjugate Matrix](https://reader031.vdocuments.us/reader031/viewer/2022020219/56649e8f5503460f94b93c79/html5/thumbnails/16.jpg)
Excellent Course of Liaoning
Example 2-9
Solution
![Page 17: Excellent Course of Liaoning Chapter 2 Matrix The Inverse of a Matrix & the Adjugate Matrix](https://reader031.vdocuments.us/reader031/viewer/2022020219/56649e8f5503460f94b93c79/html5/thumbnails/17.jpg)
Excellent Course of Liaoning
Summary
The Inverse of a Matrix A
AA 11
invertible is singular-non is 0invertible is AAAA