Cofactor Method for Inverses

• Let A = (aij) be an nxn matrix

• Recall, the co-factor Cij of element aij is:

Cij = (-1)i+j |Mij|

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

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

matrix of co-factors:

C11 C12 C1n

C21

Cn1

C22

Cn2

C2n

Cnn

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

A-1 = (matrix of co-factors)T1|A|

1|A|

C11 C21 Cn1

C12

C1n

C22

C2n

Cn2

Cnn

=

Examples

• Calculate the inverse of A = ac

bd

|M11| = d C11 = d M11 = d

Examples

• Calculate the inverse of A = ac

bd

|M12| = c C12 = -c M12 = c

Examples

• Calculate the inverse of A = ac

bd

|M21| = b C12 = -b M21 = b

Examples

• Calculate the inverse of A = ac

bd

|M22| = a C22 = a M22 = a

Examples

• Calculate the inverse of A = ac

bd

• Found that:

C22 = a C21 = -b C12 = -c C11 = d • So,

A-1 = (matrix of co-factors)T1|A|

Examples

• Calculate the inverse of A = ac

bd

• Found that:

C22 = a C21 = -b C12 = -c C11 = d • So,

A-1 = (matrix of co-factors)T1(ad-bc)

Examples

• Calculate the inverse of A = ac

bd

• Found that:

C22 = a C21 = -b C12 = -c C11 = d • So,

A-1 = 1(ad-bc)

C11

C21

C12

C22

T

Examples

• Calculate the inverse of A = ac

bd

• Found that:

C22 = a C21 = -b C12 = -c C11 = d • So,

A-1 = 1(ad-bc)

C11

C12

C21

C22

Examples

• Calculate the inverse of A = ac

bd

• Found that:

C22 = a C21 = -b C12 = -c C11 = d • So,

A-1 = 1(ad-bc)

dC12

C21

C22

Examples

• Calculate the inverse of A = ac

bd

• Found that:

C22 = a C21 = -b C12 = -c C11 = d • So,

A-1 = 1(ad-bc)

dC12

-bC22

Examples

• Calculate the inverse of A = ac

bd

• Found that:

C22 = a C21 = -b C12 = -c C11 = d • So,

A-1 = 1(ad-bc)

d-c

-bC22

Examples

• Calculate the inverse of A = ac

bd

• Found that:

C22 = a C21 = -b C12 = -c C11 = d • So,

A-1 = 1(ad-bc)

d-c

-ba

Examples 3x3 Matrix

• Calculate the inverse of B = 11

12

• Find the co-factors:

22

1

43

C11 = 2 |M11| = 2 M11 = 2 243

Examples 3x3 Matrix

• Calculate the inverse of B = 11

12

• Find the co-factors:

22

1

43

C12 = 0 |M12| = 0 M12 = 1 242

Examples 3x3 Matrix

• Calculate the inverse of B = 11

12

• Find the co-factors:

22

1

43

C13 = -1 |M13| = -1 M13 = 1 232

Examples 3x3 Matrix

• Calculate the inverse of B = 11

12

• Find the co-factors:

22

1

43

C21 = -1 |M21| = 1 M21 = 1 143

Examples 3x3 Matrix

• Calculate the inverse of B = 11

12

• Find the co-factors:

22

1

43

C22 = 2 |M22| = 2 M22 = 1 142

Examples 3x3 Matrix

• Calculate the inverse of B = 11

12

• Find the co-factors:

22

1

43

C23 = -1 |M23| = 1 M23 = 1 132

Examples 3x3 Matrix

• Calculate the inverse of B = 11

12

• Find the co-factors:

22

1

43

C31 = 0 |M31| = 0 M31 = 1 122

Examples 3x3 Matrix

• Calculate the inverse of B = 11

12

• Find the co-factors:

22

1

43

C32 = -1 |M32| = 1 M32 = 1 121

Examples 3x3 Matrix

• Calculate the inverse of B = 11

12

• First find the co-factors:

22

1

43

C33 = 1 |M33| = 1 M33 = 1 121

Examples 3x3 Matrix

• Calculate the inverse of B = 11

12

• Next the determinant: use the top row:

22

1

43

|B| = 1x |M11| -1x |M12| + 1x |M13|

= 2 – 0 + (-1) = 1

• Using the formula,

B-1 = (matrix of co-factors)T1|B|

= (matrix of co-factors)T11

Examples 3x3 Matrix

Examples 3x3 Matrix

2-1

02 -1

0

1

1-1

• Using the formula,

B-1 = (matrix of co-factors)T1|B|

= 11

T

20

-12 -1

-1

0

1-1

Examples 3x3 Matrix

• Using the formula,

B-1 = (matrix of co-factors)T1|B|

=

• Same answer obtained by Gauss-Jordan method