1. MAT 2401 Linear Algebra 3.1 The Determinant of a Matrix http://myhome.spu.edu/lauw

2. HW Written Homework

3. Preview How do I know a matrix is invertible? We will look at determinant that tells us the answer.

4. If D=ad-bc ≠ 0 the inverse of is given by Recall Therefore, if D≠0, D is called the _________ of A

5. If D=ad-bc = 0 the inverse of DNE. Fact If D=0, A is singular. To see this, for a ≠ 0, we can do the following:

6. The Task Given a square matrix A, we wish to associate with A a scalar det(A) that will tell us whether or not A is invertible

7. Fact (3.3) A square matrix A is invertible if and only if det(A)≠0

8. Interesting Comments Interesting comments from a text: The concept of determinant is subtle and not intuitive, and researchers had to accumulate a large body of experience before they were able to formulate a “correct” definition for this number.

9. n=2 1. Notations: 2. Mental picture for memorizing

10. n=3

11. Q1: What? Do I need to remember this? Q2: What if A is 4x4 or bigger? Q3: Is there a formula for 1x1 matrix?

12. Observations

15. We need: 1. a notion of “one size smaller” but related determinants. 2. a way to assign the correct signs to these smaller determinants. 3. a way to extend the computations to nxn matrices.

16. Minors and Cofactors A=[a ij ], a nxn Matrix. Let M ij be the determinant of the (n-1)x(n-1) matrix obtained from A by deleting the row and column containing a ij. M ij is called the minor of a ij. Example:

17. Minors and Cofactors A=[a ij ], a nxn Matrix. Let C ij =(-1) i+j M ij C ij is called the cofactor of a ij. Example:

18. n=3

19. Determinants Formally defined Inductively by using cofactors (minors) for all nxn matrices in a similar fashion. The process is sometimes referred as Cofactors Expansion.

20. Cofactors Expansion (across the first column) The determinant of a nxn matrix A=[a ij ] is a scalar defined by

21. Example 1

22. Remark The cofactor expansion can be done across any column or any row.

23. Sign Pattern

24. Cofactors Expansion

25. Special Matrices and Their Determinants (Square) Zero Matrix det(O)=? Identity Matrix det(I)=? We will come back to this later….

26. Upper Triangular Matrix

27. Lower Triangular Matrix

28. Diagonal Matrix

29. Q: T or F: A diagonal matrix is upper triangular?

30. Example 2

31. Determinant of a Triangular Matrix Let A=[a ij ], be a nxn Triangular Matrix, det(A)=

32. Special Matrices and Their Determinants (Square) Zero Matrix det(O)= Identity Matrix det(I)=