the inverse of the matrix

the inverse of the matrix

1. 2018 Lecture №3 the inverse of the matrix Dobroshtan О lena

2. 2 Determine the determinant of the matrix shown below.

3. 3 Determine C=AB.

4. 4

5. 5 Determine D=BA.

6. Cofactor Method for Inverses Let A = ( a ij ) be an nxn matrix Recall, the co-factor C ij of element a ij is: C ij = (-1) i+j | M ij | M ij is the (n-1) x (n-1) matrix made by removing the ROW i and COLUMN j of A

7. Cofactor Method for Inverses Put all co-factors in a matrix – called the matrix of co-factors: C 11 C 12 C 1n C 21 C n1 C 22 C n2 C 2n C nn

8. Cofactor Method for Inverses Inverse of A is given by: A -1 = (matrix of co-factors) T 1 | A | 1 | A | C 11 C 21 C n1 C 12 C 1n C 22 C 2n C n2 C nn =

9. Examples Calculate the inverse of A = a c b d | M 11 | = d C 11 = d M 11 = d

10. Examples Calculate the inverse of A = a c b d | M 12 | = c C 12 = -c M 12 = c

11. Examples Calculate the inverse of A = a c b d | M 21 | = b C 12 = -b M 21 = b

12. Examples Calculate the inverse of A = a c b d | M 22 | = a C 22 = a M 22 = a

13. Examples Calculate the inverse of A = a c b d Found that: C 22 = a C 21 = -b C 12 = -c C 11 = d So, A -1 = (matrix of co-factors) T 1 | A |

14. Examples Calculate the inverse of A = a c b d Found that: C 22 = a C 21 = -b C 12 = -c C 11 = d So, A -1 = (matrix of co-factors) T 1 (ad-bc)

15. Examples Calculate the inverse of A = a c b d Found that: C 22 = a C 21 = -b C 12 = -c C 11 = d So, A -1 = 1 (ad-bc) C 11 C 21 C 12 C 22 T

16. Examples Calculate the inverse of A = a c b d Found that: C 22 = a C 21 = -b C 12 = -c C 11 = d So, A -1 = 1 (ad-bc) C 11 C 12 C 21 C 22

17. Examples Calculate the inverse of A = a c b d Found that: C 22 = a C 21 = -b C 12 = -c C 11 = d So, A -1 = 1 (ad-bc) d C 12 C 21 C 22

18. Examples Calculate the inverse of A = a c b d Found that: C 22 = a C 21 = -b C 12 = -c C 11 = d So, A -1 = 1 (ad-bc) d C 12 -b C 22

19. Examples Calculate the inverse of A = a c b d Found that: C 22 = a C 21 = -b C 12 = -c C 11 = d So, A -1 = 1 (ad-bc) d -c -b C 22

20. Examples Calculate the inverse of A = a c b d Found that: C 22 = a C 21 = -b C 12 = -c C 11 = d So, A -1 = 1 (ad-bc) d -c -b a

21. Examples 3x3 Matrix Calculate the inverse of B = 1 1 1 2 Find the co-factors: 2 2 1 4 3 C 11 = 2 | M 11 | = 2 M 11 = 2 2 4 3

22. Examples 3x3 Matrix Calculate the inverse of B = 1 1 1 2 Find the co-factors: 2 2 1 4 3 C 12 = 0 | M 12 | = 0 M 12 = 1 2 4 2

23. Examples 3x3 Matrix Calculate the inverse of B = 1 1 1 2 Find the co-factors: 2 2 1 4 3 C 13 = -1 | M 13 | = -1 M 13 = 1 2 3 2

24. Examples 3x3 Matrix Calculate the inverse of B = 1 1 1 2 Find the co-factors: 2 2 1 4 3 C 21 = -1 | M 21 | = 1 M 21 = 1 1 4 3

25. Examples 3x3 Matrix Calculate the inverse of B = 1 1 1 2 Find the co-factors: 2 2 1 4 3 C 22 = 2 | M 22 | = 2 M 22 = 1 1 4 2

26. Examples 3x3 Matrix Calculate the inverse of B = 1 1 1 2 Find the co-factors: 2 2 1 4 3 C 23 = -1 | M 23 | = 1 M 23 = 1 1 3 2

27. Examples 3x3 Matrix Calculate the inverse of B = 1 1 1 2 Find the co-factors: 2 2 1 4 3 C 31 = 0 | M 31 | = 0 M 31 = 1 1 2 2

28. Examples 3x3 Matrix Calculate the inverse of B = 1 1 1 2 Find the co-factors: 2 2 1 4 3 C 32 = -1 | M 32 | = 1 M 32 = 1 1 2 1

29. Examples 3x3 Matrix Calculate the inverse of B = 1 1 1 2 First find the co-factors: 2 2 1 4 3 C 33 = 1 | M 33 | = 1 M 33 = 1 1 2 1

30. Examples 3x3 Matrix Calculate the inverse of B = 1 1 1 2 Next the determinant: use the top row: 2 2 1 4 3 | B | = 1x | M 11 | -1x | M 12 | + 1x | M 13 | = 2 – 0 + (-1) = 1

31. Using the formula, B -1 = (matrix of co-factors) T 1 | B | = (matrix of co-factors) T 1 1 Examples 3x3 Matrix

32. Examples 3x3 Matrix 2 -1 2 -1 1 1 -1 Using the formula, B -1 = (matrix of co-factors) T 1 | B | = 1 1 T

33. 2 -1 2 -1 -1 1 -1 Examples 3x3 Matrix Using the formula, B -1 = (matrix of co-factors) T 1 | B | = Same answer obtained by Gauss-Jordan method

34. 34 The inverse of A below is developed in the text.

35. 35 The inverse of A below is developed in the text.

36. 36 Simultaneous Linear Equations

37. 37 Matrix Form of Simultaneous Linear Equations

38. 38 Define variables as follows:

39. 39 Matrix Solution Development

40. 40 The general form and the final solution follow.

41. 41 Use matrices to solve the simultaneous equations below.

42. 42

43. 43

44. To show that matrices are inverses of one another, show that the multiplication of the matrices is commutative and results in the identity matrix. Show that A and B are inverses.

45. Use the equation AB = E Write and solve the equation: