Linear Algebra

1. Linear Algebra Chapter 3 Determinants （行列式）

2. Topics • Preliminaries（前言） • Definition（定義） • Properties of Determinants（行列式的性質） • Cofactor Expansion（代數余子式展開） • Inverse of a Matrix（矩陣的逆） • Other Applications of Determinants（行列式的應用）

3. Topics • Preliminaries • Definition • Properties of Determinants • Cofactor Expansion • Inverse of a Matrix • Other Applications of Determinants

4. Cofactor Expansion • Comment - To this point, there has been only one way to calculate the determinant, i.e. from the definition There are other ways of calculating the determinant that can be useful for calculations and for proofs

5. Cofactor Expansion • Defn - Let A= [ aij ] be an n × n matrix. Let Mpq be the (n – 1) × (n – 1) submatrix of A obtained by deleting the pth row and qth column of A. The determinant det(Mpq) is called the minor of apq(apq的子式)。 • Defn - Let A= [ aij ] be an n × n matrix. The cofactorApq of apq（ apq 的余子式）is defined as Apq= (–1)p+q det(Mpq)

6. Cofactor Expansion Example • Let

7. Exercise Let ， Find the cofactors of a11 and a32

8. Cofactor Expansion Comment • Examine pattern of signs of term (–1) p+q When using cofactors, don’t have to evaluate (–1) p+qJust remember the pattern n = 3 n = 4

9. Cofactor Expansion • Theorem - Let A= [ aij ] be an n × n matrix. Then det(A) =ai1Ai1+ ai2Ai2 + …+ ain Ain (expansion of det(A) with respect to row i ) and det(A) =a1jA1j+ a2jA2j + … + anj Anj (expansion of det(A) with respect to column j )

10. Cofactor Expansion Example • Expand the determinant of a 3 × 3 matrix det(A) =a11a22a33 + a12a23a31 + a13a21a32 – a11a23a32 – a12a21a33 – a13a22a31 Reorganize expression with respect to first row det(A) =a11(a22a33 – a23a32) + a12(a23a31 – a21a33) + a13(a21a32 – a22a31)

11. Cofactor Expansion Example (continued) Try to recognize the terms in parentheses So det(A) =a11A11 + a12A12 + a13A13

12. Cofactor Expansion Example (continued) Regroup det(A) with respect to the first column of A det(A) =a11(a22a33 – a23a32) +a21(a13a32 – a12a33 ) + a31(a12a23 – a13a22) So det(A) =a11A11 + a21A21 + a31A31

13. Example: Evaluate the determinant It is better to expand along the column 2.

14. Cofactor Expansion Example • Evaluate Pick row or column with large number of zeros, such as column 2 (or row 3)

15. Comment-we can use the properties of determinant to introduce many zeros in a given row or column and then expand along that row or column. Example

16. Exercise: Evaluate the following determinants.

17. Topics • Preliminaries • Definition • Properties of Determinants • Cofactor Expansion • Inverse of a Matrix • Other Applications of Determinants

18. Inverse of a Matrix • Have seen an explicit formula for inverse of a 2 × 2 matrix In terms of more general notation this is

19. Inverse of a Matrix • Theorem - Let A= [ aij ] be an n × n matrix. Then ai1Ak1 + ai2Ak2 + … + ainAkn= 0 for i ≠ ka1jA1k + a2jA2k + … + anjAnk = 0 for j ≠ k i.e. sum of row or column times cofactors of another row or column is 0.

20. Inverse of a Matrix Example • Let

21. Inverse of a Matrix • Defn - Let A= [ aij ] be an n × n matrix. The matrix is called the transpose of the matrix of cofactors（余因子矩阵的转置）or (book’s term) adjoint（伴随矩阵）of A, notation adj(A)

22. Inverse of a Matrix Example • Let

23. Inverse of a Matrix Example (continued)

24. adj(A)= Inverse of a Matrix • Theorem - Let A= [ aij ] be an n × n matrix. Then A( adj(A) ) = ( adj(A) )A= det(A)In Example

25. Inverse of a Matrix • Theorem - Let A be an n × n matrix with det(A) ≠ 0. Then

26. Inverse of a Matrix Example

27. Exercise:

28. Topics • Preliminaries • Definition • Properties of Determinants • Cofactor Expansion • Inverse of a Matrix • Other Applications of Determinants

29. Other Applications of Determinants • Theorem - (Cramer’s Rule) Consider the linear system of n equations in n unknowns Let A= [ aij ] be the coefficient matrix, so that the system may be expressed as Ax=b. If det(A)≠0, then the system has a unique solution Where Ai is the matrix obtained from A by replacing its ith column with b

30. Other Applications of Determinants • Proof - If det(A) ≠ 0, then A is nonsingular. So

31. Other Applications of Determinants • Proof (continued) Define Ai as A with the ith column replaced by b Evaluate det(Ai) by expanding with respect to ith column, then So

32. Other Applications of Determinants Example • Solve the system

33. Exercise: Solve the system

34. Computing Determinants Comment- • It is practical to use determinants for computations involving an n×n matrix A when n ≤ 4 • For larger systems, other methods such as Gaussian elimination run much more quickly