1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 12 System of Linear Equations

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

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

5. Matrix Algebra 3 rd Row 2 nd Column

6. Matrix Algebra 1 Row, m Columns Row Vector

7. Matrix Algebra n Rows, 1 Column Column Vector

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

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

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

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

12. Matrix Algebra Upper Triangular Special Types of Square Matrices

13. Matrix Algebra Lower Triangular Special Types of Square Matrices

14. Matrix Algebra Banded Special Types of Square Matrices

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

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

17. Matrix Operating Rules - Addition Properties [A]+[B] = [B]+[A] [A]+([B]+[C]) = ([A]+[B])+[C]

18. Multiplication by Scalar

19. Matrix Multiplication [A] n x m. [B] p x q = [C] n x q m=p

20. Matrix Multiplication

22. Example

23. Matrix Multiplication - Properties Associative: [A]([B][C]) = ([A][B])[C] If dimensions suitable Distributive: [A]([B]+[C]) = [A][B]+[A] [C] Attention: [A][B]  [B][A]

24. Operations - Transpose

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

26. Operations - Trace Square Matrix tr[A] =  a ii

27. Linear Equations in Matrix Form

32. Homework Problems 9.1, 9.2, 9.3 Due Date: Oct 6