MAT 2401 Linear Algebra

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

2. HW • WebAssign 3.1 • Written Homework

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

4. Recall If D=ad-bc ≠ 0 the inverse of is given by

5. Fact If D=ad-bc = 0 the inverse of DNE.

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

10. n=3

11. n=3

12. Observations

13. Observations

14. Observations

15. Observations

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

17. Minors and Cofactors A=[aij], a nxn Matrix. Let Cij =(-1)i+jMij Cij is called the cofactor of aij.

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=[aij] is a scalar defined by

21. Example 1

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

23. Cofactors Expansion

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

25. Upper Triangular Matrix

26. Lower Triangular Matrix

27. Diagonal Matrix

28. Diagonal Matrix Q: T or F: A diagonal matrix is upper triangular?

29. Example 2

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

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