1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 14 Elimination Methods

2. Objectives Introduction to Matrix Algebra Express System of Equations in Matrix Form Introduce Methods for Solving Systems of Equations Advantages and Disadvantages of each Method

3. Last Time Matrix Algebra Rectangular Array of Elements Represented by a single symbol [A]

4. Last Time Matrix Algebra Row 1 Row 3 Column 2Column m n x m Matrix

5. Last Time Matrix Algebra 3 rd Row 2 nd Column

6. Last Time Matrix Algebra 1 Row, m Columns Row Vector

7. Last Time Matrix Algebra n Rows, 1 Column Column Vector

8. Last Time Matrix Algebra If n = m Square Matrix e.g. n=m=5 Main Diagonal

9. Last Time Matrix Algebra Special Types of Square Matrices Symmetric: a ij = a ji

10. Last Time Matrix Algebra Diagonal: a ij = 0, i  j Special Types of Square Matrices

11. Last Time Matrix Algebra Identity: a ii =1.0 a ij = 0, i  j Special Types of Square Matrices

12. Last Time Matrix Algebra Upper Triangular Special Types of Square Matrices

13. Last Time Matrix Algebra Lower Triangular Special Types of Square Matrices

14. Last Time Matrix Algebra Banded Special Types of Square Matrices

15. Last Time Matrix Operating Rules - Equality [A] mxn =[B] pxq n=pm=qa ij =b ij

16. Last Time Matrix Operating Rules - Addition [C] mxn = [A] mxn +[B] pxq n=p m=q c ij = a ij +b ij

17. Last Time Multiplication by Scalar

18. Last Time Matrix Multiplication [A] n x m. [B] p x q = [C] n x q m=p

19. Last Time Matrix Multiplication

21. Last Time Operations - Transpose

22. Last Time Operations - Inverse [A][A] -1 [A] [A] -1 =[I] If [A] -1 does not exist [A] is singular

23. Last Time Operations - Trace Square Matrix tr[A] =  a ii

24. Equations in Matrix Form Consider

25. Linear Equations in Matrix Form

31. # Equations = # Unknowns = n Square Matrix n x n

32. Solution of Linear Equations Consider the system

33. Solution of Linear Equations

36. Express In Matrix Form Upper Triangular What is the characteristic? Solution by Back Substitution

37. Solution of Linear Equations Objective Can we express any system of equations in a form 0

38. Background Consider (Eq 1) (Eq 2) Solution 2*(Eq 1) (Eq 2) Solution !!!!!! Scaling Does Not Change the Solution

39. Background Consider (Eq 1) (Eq 2)-(Eq 1) Solution !!!!!! (Eq 1) (Eq 2) Solution Operations Do Not Change the Solution

40. Gauss Elimination Example Forward Elimination

41. Gauss Elimination -

42. Substitute 2 nd eq with new

43. Gauss Elimination -

44. Substitute 3rd eq with new

45. Gauss Elimination -

46. Substitute 3rd eq with new

47. Gauss Elimination