1. ME 142 Engineering Computation I Matrix Operations in Excel

2. Key Concepts Matrix Basics Matrix Multiplication Transposing a Matrix Inverting a Matrix Determinant of a Matrix Cramer’s Rule Matrix Inversion Method

3. Matrix Basics

4. What is a Matrix?

5. Matrix Basics  How are Matrices useful in engineering?

6. Matrix Basics  Used named cells to define arrays Simplifies process Makes formulas easier to understand  Pre-select the array output area  Use [Shift]+[Ctrl]+[Enter] key combination to execute array commands Populates each cell in array output area with array command

7. Matrix Addition

8. =A+B [Shift]+[Cntl]+[Enter]

9. Matrix Multiplication

10. How do you multiply Matrices?

11. Example Problem  Given matrices A and B, A x B equals:

12. Matrix Multiplication: AxB =MMULT(A,B) [Shift]+[Cntl]+[Enter]

13. Matrix Multiplication: BxA =MMULT(A,B) [Shift]+[Cntl]+[Enter]

14. Transposing a Matrix

15. How do you transpose a matrix? Can any matrix be transposed?

16. Transposing a Matrix =TRANSPOSE(A) [Shift]+[Cntl]+[Enter]

17. Inverting a Matrix

18. =MINVERSE(D) [Shift]+[Cntl]+[Enter]

19. Inverting a Matrix

20. Can any matrix be inverted?

21. Determinant of a Matrix

22. How do you calculate the determinant of a matrix?

23. Example Problem  Find the determinant of the matrix:

24. Determinant of a Matrix =MDETERM(D) D determinant =211

25. Can you calculate the determinant of any matrix?

26. Cramer’s Rule

27.  Useful in solving systems of 2 or 3 linear equations, by hand or by computer  This rule states that each unknown in a system of linear equations may be expressed as a fraction of two determinants. The determinant of the denominator, D, is obtained from the coefficients of matrix [A] The determinant of the numerator is obtained from D by replacing the column of coefficients of the unknown in question by the coefficients of matrix [B]

28. Cramer’s Rule Given linear system of equations in matrix form: Where

29. Cramer’s Rule Then the determinant of [A] may be defined as: And values of [X] may be found from the expressions below:

30. Example Problem  Use Cramer’s rule to manually solve the following: 2x + 3y = 13 5x - 1y = 7

31. Example Problem  Use Cramer’s rule to solve the following, with the assistance of Excel: 5x + 3y +1z = 2 2x + 4y + 2z = -5 4x - 3y + 6z = 3

32. Matrix Inversion Method

33. Given linear system of equations in matrix form: Where Then multiplying both sides by [A -1 ], the inversion of [A]

34. Example Problem  Use the Matrix Inversion method to solve the following set of equation: 5x + 3y +1z = 2 2x + 4y + 2z = -5 4x - 3y + 6z = 3

35. Review Questions

36. Review Question Matrix Basics A range of cells may be named, and used as an argument in a matrix function. A.True B.False

37. Review Question Matrix Multiplication Given matrices A and B, A x B equals: A. B.

38. ReviewQuestion Inverting a Matrix Can any square matrix be inverted? A.Yes B.No

39. Review Question Determinant of a Matrix Find the determinant of the matrix: A.16 B.-29 C.29 D.None of the above

40. Review Question Cramer’s Rule vs. Matrix Inversion Method Given a set of 4 linear equations to solve simultaneously, which of the following methods is computationally more efficient: A.Cramer’s Rule B.Matrix Inversion Method C.Either, essentially the same D.Cannot tell from information given

41. Homework Help ‘n Hints