Cofactor%20Method%20for%20Inverses - PowerPoint PPT Presentation

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Cofactor%20Method%20for%20Inverses

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Cofactor Method for Inverses Put all co-factors in a matrix called the matrix of co-factors: ... = Same answer obtained by Gauss-Jordan method . Title ... – PowerPoint PPT presentation

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Title: Cofactor%20Method%20for%20Inverses


1
Cofactor Method for Inverses
  • Let A (aij) be an nxn matrix
  • Recall, the co-factor Cij of element aij is

Cij (-1)ij Mij
  • Mij is the (n-1) x (n-1) matrix made by removing
    the ROW i and COLUMN j of A

2
Cofactor Method for Inverses
  • Put all co-factors in a matrix called the
    matrix of co-factors

3
Cofactor Method for Inverses
  • Inverse of A is given by

4
Examples
  • Calculate the inverse of A

M11 d
C11 d
5
Examples
  • Calculate the inverse of A

M12 c
C12 -c
6
Examples
  • Calculate the inverse of A

7
Examples
  • Calculate the inverse of A

8
Examples
  • Calculate the inverse of A
  • Found that

C22 a
C21 -b
C12 -c
C11 d
9
Examples
  • Calculate the inverse of A
  • Found that

C22 a
C21 -b
C12 -c
C11 d
  • So,

A-1 (matrix of co-factors)T
10
Examples
  • Calculate the inverse of A
  • Found that

C22 a
C21 -b
C12 -c
C11 d
  • So,

T
A-1
11
Examples
  • Calculate the inverse of A
  • Found that

C22 a
C21 -b
C12 -c
C11 d
  • So,

A-1
12
Examples
  • Calculate the inverse of A
  • Found that

C22 a
C21 -b
C12 -c
C11 d
  • So,

A-1
13
Examples
  • Calculate the inverse of A
  • Found that

C22 a
C21 -b
C12 -c
C11 d
  • So,

A-1
14
Examples
  • Calculate the inverse of A
  • Found that

C22 a
C21 -b
C12 -c
C11 d
  • So,

A-1
15
Examples
  • Calculate the inverse of A
  • Found that

C22 a
C21 -b
C12 -c
C11 d
  • So,

A-1
16
Examples 3x3 Matrix
1
1
1
  • Calculate the inverse of B

1
2
2
2
4
3
  • Find the co-factors

17
Examples 3x3 Matrix
1
1
1
  • Calculate the inverse of B

1
2
2
2
4
3
  • Find the co-factors

18
Examples 3x3 Matrix
1
1
1
  • Calculate the inverse of B

1
2
2
2
4
3
  • Find the co-factors

19
Examples 3x3 Matrix
1
1
1
  • Calculate the inverse of B

1
2
2
2
4
3
  • Find the co-factors

20
Examples 3x3 Matrix
1
1
1
  • Calculate the inverse of B

1
2
2
2
4
3
  • Find the co-factors

21
Examples 3x3 Matrix
1
1
1
  • Calculate the inverse of B

1
2
2
2
4
3
  • Find the co-factors

22
Examples 3x3 Matrix
1
1
1
  • Calculate the inverse of B

1
2
2
2
4
3
  • Find the co-factors

23
Examples 3x3 Matrix
1
1
1
  • Calculate the inverse of B

1
2
2
2
4
3
  • Find the co-factors

24
Examples 3x3 Matrix
1
1
1
  • Calculate the inverse of B

1
2
2
2
4
3
  • First find the co-factors

25
Examples 3x3 Matrix
1
1
1
  • Calculate the inverse of B

1
2
2
2
4
3
  • Next the determinant use the top row

B 1x M11 -1x M12 1x M13
2 0 (-1) 1
26
Examples 3x3 Matrix
27
Examples 3x3 Matrix
T
28
Examples 3x3 Matrix
  • Same answer obtained by Gauss-Jordan method
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