Download - Co-factor matrix
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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
![Page 2: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/2.jpg)
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
![Page 3: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/3.jpg)
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
=
![Page 4: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/4.jpg)
Examples
• Calculate the inverse of A = ac
bd
|M11| = d C11 = d M11 = d
![Page 5: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/5.jpg)
Examples
• Calculate the inverse of A = ac
bd
|M12| = c C12 = -c M12 = c
![Page 6: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/6.jpg)
Examples
• Calculate the inverse of A = ac
bd
|M21| = b C12 = -b M21 = b
![Page 7: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/7.jpg)
Examples
• Calculate the inverse of A = ac
bd
|M22| = a C22 = a M22 = a
![Page 8: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/8.jpg)
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|
![Page 9: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/9.jpg)
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)
![Page 10: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/10.jpg)
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
![Page 11: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/11.jpg)
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
![Page 12: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/12.jpg)
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
![Page 13: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/13.jpg)
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
![Page 14: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/14.jpg)
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
![Page 15: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/15.jpg)
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
![Page 16: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/16.jpg)
Examples 3x3 Matrix
• Calculate the inverse of B = 11
12
• Find the co-factors:
22
1
43
C11 = 2 |M11| = 2 M11 = 2 243
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Examples 3x3 Matrix
• Calculate the inverse of B = 11
12
• Find the co-factors:
22
1
43
C12 = 0 |M12| = 0 M12 = 1 242
![Page 18: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/18.jpg)
Examples 3x3 Matrix
• Calculate the inverse of B = 11
12
• Find the co-factors:
22
1
43
C13 = -1 |M13| = -1 M13 = 1 232
![Page 19: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/19.jpg)
Examples 3x3 Matrix
• Calculate the inverse of B = 11
12
• Find the co-factors:
22
1
43
C21 = -1 |M21| = 1 M21 = 1 143
![Page 20: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/20.jpg)
Examples 3x3 Matrix
• Calculate the inverse of B = 11
12
• Find the co-factors:
22
1
43
C22 = 2 |M22| = 2 M22 = 1 142
![Page 21: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/21.jpg)
Examples 3x3 Matrix
• Calculate the inverse of B = 11
12
• Find the co-factors:
22
1
43
C23 = -1 |M23| = 1 M23 = 1 132
![Page 22: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/22.jpg)
Examples 3x3 Matrix
• Calculate the inverse of B = 11
12
• Find the co-factors:
22
1
43
C31 = 0 |M31| = 0 M31 = 1 122
![Page 23: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/23.jpg)
Examples 3x3 Matrix
• Calculate the inverse of B = 11
12
• Find the co-factors:
22
1
43
C32 = -1 |M32| = 1 M32 = 1 121
![Page 24: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/24.jpg)
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
![Page 25: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/25.jpg)
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
![Page 26: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/26.jpg)
• Using the formula,
B-1 = (matrix of co-factors)T1|B|
= (matrix of co-factors)T11
Examples 3x3 Matrix
![Page 27: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/27.jpg)
Examples 3x3 Matrix
2-1
02 -1
0
1
1-1
• Using the formula,
B-1 = (matrix of co-factors)T1|B|
= 11
T
![Page 28: Co-factor matrix](https://reader036.vdocuments.mx/reader036/viewer/2022070509/58a4db711a28ab34318b5d09/html5/thumbnails/28.jpg)
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