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Excellent Course of Liaoning
Chapter 2 Matrix
The Inverse of a Matrix & the Adjugate Matrix
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.
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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