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Chapter 2 Matrix

The Inverse of a Matrix & the Adjugate Matrix

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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.

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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

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2.3.2 Adjugate Matrix and a Formula to Find InverseDefinition 2.10

!!!

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Find the adjuagte (classical adjoint) of AExample 2-5

a bA

c d

A

- cd - b

a

Solution

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Theorem 2.3

Proposition 2.1

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Proof

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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

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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

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Example 2-7

Proof

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Theorem 2.4

Proof

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Example 2-8

Proof

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Definition 2.11

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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

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2.3.4 Formulas for Adjoint (Adjugate) Matrices

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Example 2-9

Solution

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Summary

The Inverse of a Matrix A

AA 11

invertible is singular-non is 0invertible is AAAA