Co-factor matrix..
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The document describes the cofactor method for finding the inverse of a matrix A. It defines the cofactor Cij as the signed determinant of the matrix made by removing row i and column j from A. The inverse is then given by the transpose of the matrix of cofactors divided by the determinant of A. An example calculates the inverse of the 2x2 matrix A = [a, c; b, d] using this method.
Co-factor matrix..
- 1. 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
- 2. 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
- 3. 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 =
- 4. Examples • Calculate the inverse of A = a c b d |M11| = d C11 = dM11 = d
- 5. Examples • Calculate the inverse of A = a c b d |M12| = c C12 = -cM12 = c
- 6. Examples • Calculate the inverse of A = a c b d |M21| = b C12 = -bM21 = b
- 7. Examples • Calculate the inverse of A = a c b d |M22| = a C22 = aM22 = a
- 8. Examples • Calculate the inverse of A = a c b d • Found that: C22 = aC21 = -bC12 = -cC11 = d • So, A-1 = (matrix of co-factors)T1 |A|
- 9. Examples • Calculate the inverse of A = a c b d • Found that: C22 = aC21 = -bC12 = -cC11 = d • So, A-1 = (matrix of co-factors)T1 (ad-bc)
- 10. Examples • Calculate the inverse of A = a c b d • Found that: C22 = aC21 = -bC12 = -cC11 = d • So, A-1 = 1 (ad-bc) C11 C21 C12 C22 T
- 11. Examples • Calculate the inverse of A = a c b d • Found that: C22 = aC21 = -bC12 = -cC11 = d • So, A-1 = 1 (ad-bc) C11 C12 C21 C22
- 12. Examples • Calculate the inverse of A = a c b d • Found that: C22 = aC21 = -bC12 = -cC11 = d • So, A-1 = 1 (ad-bc) d C12 C21 C22
- 13. Examples • Calculate the inverse of A = a c b d • Found that: C22 = aC21 = -bC12 = -cC11 = d • So, A-1 = 1 (ad-bc) d C12 -b C22
- 14. Examples • Calculate the inverse of A = a c b d • Found that: C22 = aC21 = -bC12 = -cC11 = d • So, A-1 = 1 (ad-bc) d -c -b C22
- 15. Examples • Calculate the inverse of A = a c b d • Found that: C22 = aC21 = -bC12 = -cC11 = d • So, A-1 = 1 (ad-bc) d -c -b a
- 16. Examples 3x3 Matrix • Calculate the inverse of B = 1 1 1 2 • Find the co-factors: 2 2 1 43 C11 = 2|M11| = 2M11 = 2 2 43
- 17. Examples 3x3 Matrix • Calculate the inverse of B = 1 1 1 2 • Find the co-factors: 2 2 1 43 C12 = 0|M12| = 0M12 = 1 2 42
- 18. Examples 3x3 Matrix • Calculate the inverse of B = 1 1 1 2 • Find the co-factors: 2 2 1 43 C13 = -1|M13| = -1M13 = 1 2 32
- 19. Examples 3x3 Matrix • Calculate the inverse of B = 1 1 1 2 • Find the co-factors: 2 2 1 43 C21 = -1|M21| = 1M21 = 1 1 43
- 20. Examples 3x3 Matrix • Calculate the inverse of B = 1 1 1 2 • Find the co-factors: 2 2 1 43 C22 = 2|M22| = 2M22 = 1 1 42
- 21. Examples 3x3 Matrix • Calculate the inverse of B = 1 1 1 2 • Find the co-factors: 2 2 1 43 C23 = -1|M23| = 1M23 = 1 1 32
- 22. Examples 3x3 Matrix • Calculate the inverse of B = 1 1 1 2 • Find the co-factors: 2 2 1 43 C31 = 0|M31| = 0M31 = 1 1 22
- 23. Examples 3x3 Matrix • Calculate the inverse of B = 1 1 1 2 • Find the co-factors: 2 2 1 43 C32 = -1|M32| = 1M32 = 1 1 21
- 24. Examples 3x3 Matrix • Calculate the inverse of B = 1 1 1 2 • First find the co-factors: 2 2 1 43 C33 = 1|M33| = 1M33 = 1 1 21
- 25. Examples 3x3 Matrix • Calculate the inverse of B = 1 1 1 2 • Next the determinant: use the top row: 2 2 1 43 |B| = 1x |M11| -1x |M12| + 1x |M13| = 2 – 0 + (-1) = 1
- 26. • Using the formula, B-1 = (matrix of co-factors)T1 |B| = (matrix of co-factors)T1 1 Examples 3x3 Matrix
- 27. Examples 3x3 Matrix 2 -1 0 2 -1 0 1 1-1 • Using the formula, B-1 = (matrix of co-factors)T1 |B| = 1 1 T
- 28. 2 0 -1 2 -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