Introducing the Determinant

1. C. Ray Rosentrater Westmont College Introducing the Determinant rosentr@westmont.edu 2013 Joint Mathematics Meetings

2. When students are introduced to a new concept via a problem they understand: They can be engaged in exploratory/active learning exercises. They understand the new concept better. They are more willing to engage in theoretical analysis of the concept. Premise

3. What are the common approaches?

4. Motivation: Want to study a function with a matrix variable. Development Thread: Permutations Elementary Products (Definition) Evaluation by Row Reduction (No justification) Properties Cofactor Expansion (No justifiction) Application: Crammer’s Rule Presentation of the determinant: Text 1

5. Motivation: Another important number associated with a square matrix. Development Thread: Permutations Definition Properties (Row ops & Evaluation via triangular matrices) Computation via Cofactors (3x3 justified) Applications: Crammer’s rule Presentation of the determinant: Text 2

6. Motivation: List of uses (Singularity test, Volume, Sensitivity analysis) Development Thread: • Properties • Identity matrix, row exchange, linear in row one • Zero row, duplicate rows, triangular matrices, product rule, transpose (proved from first set) • Computation: Permutations and Cofactors • Applications: Cramer’s rule, Volume Presentation of the determinant: Text 3

7. Motivation: Associate a real number to a matrix A in such a way that we can tell if A is singular. Development Thread: 2x2, 3x3 singularity testing Cofactor Definition Properties (Row operations, Product) Applications: Crammer’s rule, Matrix codes, Cross product Presentation of the determinant: Text 4

8. Motivation: Singularity testing Development Thread: 2x2, 3x3 singularity testing Cofactor Definition Properties: Row operations (not justified), Products, Transposes Applications: Crammer’s rule, Volume, Transformations Presentation of the determinant: Text 5

9. Singularity Checking (Text 5) E. G. O. Not amenable to active learning

10. Motivation: Signed Area/Volume/Hyper-volume of the parallelogram (etc.) spanned by the rows Development Thread: Simple Cases Row operations Semi-formal definition & computational method Transition to Cofactor (Permutation) Definition Properties Proposed Presentation of the determinant

11. Simple Cases

12. Simple cases

13. Motivation: Signed Area/Volume/Hyper-volume spanned by the rows Development Thread: Simple Cases Row operations Semi-formal definition Transition to Cofactor (Permutation) Definition Properties Proposed Presentation of the determinant

14. Row operations: Row Scaling

15. Row operations: Row swap

16. Row operations: Row replacement

17. Motivation: Signed Area/Volume/Hyper-volume spanned by the rows Development Thread: Simple Cases Row operations Semi-formal definition & computation Transition to Cofactor (Permutation) Definition Properties Proposed Presentation of the determinant

18. Determinant = signed “volume” of the parallelogram spanned by the rows To Compute: Use row replacements to put in triangular form, multiply the diagonal entries First Definition

19. Motivation: Signed Area/Volume/Hyper-volume spanned by the rows Development Thread: Simple Cases Row operations Semi-formal definition Transition to Cofactor (Permutation) Definition Properties Proposed Presentation of the determinant

20. Transition to Cofactor definition Why have another method? A motivating example

21. Transition to Cofactor definition Why have another method? A motivating example

22. Transition to Cofactor definition

23. Transition to Cofactor definition • State the Cofactor definition • Verify the definitions agree • Check simple (diagonal) case • Check row operation behavior

24. Verify scaling in row one from definition • To scale row k • Swap row k with row one • Scale row one • Swap row k and row one Row scaling:

25. Row replacement: • To add a multiple of row k to row j: • Swap rows one and j • Add the multiple of row k to row one • Swap rows one and j

26. swapping first two rows A B

27. Interchanging first two rows A B

28. Induction • If the first row is not involved, use the inductive hypothesis • If the first row is to be swapped with row k, • Swap row k with row two • Swap rows one and two • Swap row k with row two Swapping other rows:

29. Motivation: Signed Area/Volume/Hyper-volume spanned by the rows Development Thread: Simple Cases Row operations Semi-formal definition Transition to Cofactor (Permutation) Definition Properties Proposed Presentation of the determinant

30. Better motivation Multiple views Students can develop significant ideas on their own: Active Learning Students can anticipate theoretical ideas Students are motivated to prove row operation results Benefits of a Volume-first approach Thank you rosentr@westmont.edu

31. Associated materials may be obtained by contacting Ray Rosentrater Westmont College 955 La Paz Rd Santa Barbara, CA 93108 805.565.6185 rosentr@westmont.edu