1. ALGEBRA AND TRIGONOMETRYBy
DR. INDU JINDAL
Deptt. of Maths
PGGCG SEC 42 CHD

2. Elementary Operations and Rank Of A Matrix

3. Some Definitions
A matrix A is said to be a row (column) equivalent to a matrix B if B can be obtained from A after a finite number of elementary row(column) operations i.e. A∼ B.

5. A matrix in echelon form is said to be in the row reduced echelon form if
Distinguished elements are equal to 1,and
The column which contains the distinguished element has all other elements equal to zero.
Remark : Every matrix A is row equivalent to a matrix in the echelon form or row reduced echelon form.

6. A matrix obtained from an identity matrix , by subjecting it to an elementary operation is called an elementary matrix. It is written as E-matrix.
Theorem: Each elementary row (column) operation on a matrix A has the same effect on A as the pre-multiplication (post-multiplication) of A by the corresponding elementary matrix.
Lemma: Every elementary row (column) operation on the product of two matrices can be affected by subjecting the pre-factor (post-factor) of the product of the same row (column) operation.

7. Types of elementary matrices

8. Inverse of a Matrix

9. Inverse using elementary operations

10. Example 1

11. Example 2

12. Example 3

13. Example 4

14. Example 5

15. Example 6

16. Example 7

17. Example 8

18. Example 9
Prove that a skew-symmetric matrix of odd order cannot be invertible.

19. Example 10

20. Minor of a matrix

21. Example 11
Prove that the inverse of a non-singular symmetric matrix is symmetric.

22. Rank

23. Theorems

24. Example 1

25. Example 2

26. Example 3

27. Example 4

28. Example 5

29. Example 6

30. Rank of matrix using Elementary Operations

32. Example 1

33. Example 2

34. Row and Column Vectors

35. Linearly dependent

36. Linearly independent

37. Example 1

38. Example 2

39. Example 3

40. Example 4

41. Row-rank and Column-rank

42. Example

43. Equivalence of matrices
Two matrices A and B of same order over a field F are said to be equivalent if there exist non-singular matrices P and Q over F such that B=PAQ Theorem The row rank, the column rank and rank of a matrix are equal.

44. Example 1

45. Example 2

46. Example 3

47. Example 4

48. Example 5

49. Example 6

50. Example 7

51. Example 8
Show that if two matrices A and B over same field are of same type and have same rank, then they are equivalent matrices.