Transcript The Cofactor Method for Inverting a Matrix

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
C21 C22
C1n
C2n
Cn1 Cn2
Cnn
Cofactor Method for Inverses
• Inverse of A is given by:
A-1
1
=
(matrix of co-factors)T
|A|
1
= |A|
C11 C21
C12 C22
Cn1
Cn2
C1n C2n
Cnn
Examples
• Calculate the inverse of A =
M11 = d
|M11| = d
a
b
c
d
C11 = d
Examples
• Calculate the inverse of A =
M12 = c
|M12| = c
a
b
c
d
C12 = -c
Examples
• Calculate the inverse of A =
M21 = b
|M21| = b
a
b
c
d
C12 = -b
Examples
• Calculate the inverse of A =
M22 = a
|M22| = a
a
b
c
d
C22 = a
Examples
• Calculate the inverse of A =
a
b
c
d
• Found that:
C11 = d
C12 = -c
C21 = -b C22 = a
• So,
A-1
1
=
(matrix of co-factors)T
|A|
Examples
• Calculate the inverse of A =
a
b
c
d
• Found that:
C11 = d
C12 = -c
C21 = -b C22 = a
• So,
A-1
1
=
(matrix of co-factors)T
(ad-bc)
Examples
• Calculate the inverse of A =
a
b
c
d
• Found that:
C11 = d
C12 = -c
C21 = -b C22 = a
• So,
C11 C12
1
A-1 =
(ad-bc) C C
21
22
T
Examples
• Calculate the inverse of A =
a
b
c
d
• Found that:
C11 = d
C12 = -c
C21 = -b C22 = a
• So,
C11 C21
1
A-1 =
(ad-bc) C C
12
22
Examples
• Calculate the inverse of A =
a
b
c
d
• Found that:
C11 = d
C12 = -c
C21 = -b C22 = a
• So,
C21
d
1
A-1 =
(ad-bc) C C
12
22
Examples
• Calculate the inverse of A =
a
b
c
d
• Found that:
C11 = d
C12 = -c
C21 = -b C22 = a
• So,
A-1
d -b
1
=
(ad-bc) C C
12
22
Examples
• Calculate the inverse of A =
a
b
c
d
• Found that:
C11 = d
C12 = -c
C21 = -b C22 = a
• So,
A-1
d -b
1
=
(ad-bc) -c C
22
Examples
• Calculate the inverse of A =
a
b
c
d
• Found that:
C11 = d
C12 = -c
C21 = -b C22 = a
• So,
A-1
d -b
1
=
(ad-bc) -c a
Examples 3x3 Matrix
1
• Calculate the inverse of B = 1
2
1 1
2 2
3 4
• Find the co-factors:
2
2
M11 =
3 4
|M11| = 2
C11 = 2
Examples 3x3 Matrix
1
• Calculate the inverse of B = 1
2
1 1
2 2
3 4
• Find the co-factors:
1
2
M12 =
2 4
|M12| = 0
C12 = 0
Examples 3x3 Matrix
1
• Calculate the inverse of B = 1
2
1 1
2 2
3 4
• Find the co-factors:
1
2
M13 =
2 3
|M13| = -1
C13 = -1
Examples 3x3 Matrix
1
• Calculate the inverse of B = 1
2
1 1
2 2
3 4
• Find the co-factors:
1
1
M21 =
3 4
|M21| = 1
C21 = -1
Examples 3x3 Matrix
1
• Calculate the inverse of B = 1
2
1 1
2 2
3 4
• Find the co-factors:
1
1
M22 =
2 4
|M22| = 2
C22 = 2
Examples 3x3 Matrix
1
• Calculate the inverse of B = 1
2
1 1
2 2
3 4
• Find the co-factors:
1
1
M23 =
2 3
|M23| = 1
C23 = -1
Examples 3x3 Matrix
1
• Calculate the inverse of B = 1
2
1 1
2 2
3 4
• Find the co-factors:
1
1
M31 =
2 2
|M31| = 0
C31 = 0
Examples 3x3 Matrix
1
• Calculate the inverse of B = 1
2
1 1
2 2
3 4
• Find the co-factors:
1
1
M32 =
1 2
|M32| = 1
C32 = -1
Examples 3x3 Matrix
1
• Calculate the inverse of B = 1
2
1 1
2 2
3 4
• First find the co-factors:
1
1
M33 =
1 2
|M33| = 1
C33 = 1
Examples 3x3 Matrix
1
• Calculate the inverse of B = 1
2
1 1
2 2
3 4
• Next the determinant: use the top row:
|B| = 1x |M11| -1x |M12| + 1x |M13|
= 2 – 0 + (-1) = 1
Examples 3x3 Matrix
• Using the formula,
B-1
1
=
(matrix of co-factors)T
|B|
1
=
(matrix of co-factors)T
1
Examples 3x3 Matrix
• Using the formula,
B-1
1
=
(matrix of co-factors)T
|B|
1
2
0
1
=
1 -1 2 -1
0 -1 1
T
Examples 3x3 Matrix
• Using the formula,
B-1
1
=
(matrix of co-factors)T
|B|
=
2 -1 0
0 2 -1
-1 -1 1
• Same answer obtained by Gauss-Jordan method