Cofactor Method for Inverses

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