1. 2 - 1 Chapter 2A Matrices 2A.1 Definition, and Operations of Matrices: 1 Sums and Scalar Products; 2 Matrix Multiplication 2A.2 Properties of Matrix Operations; Mechanics of Matrix Multiplication 2A.3 The Inverse of a Matrix 2A.4 Elementary Matrices and LU factorization of a matrix

2. 2 - 2 2A.1 Operations with Matrices Matrix: (i, j)-th entry: row: m column: n size: m×n

3. 2 - 3 i-th row vector j-th column vector row matrix column matrix Square matrix: m = n

4. 2 - 4 Diagonal matrix: Trace:

5. 2 - 5 Ex:

6. 2 - 6 Partitioned matrices: submatrix

7. 2 - 7 Equal matrix:

8. 2 - 8 Matrix addition: Scalar multiplication:

9. 2 - 9 Matrix subtraction: Example: Matrix addition

10. 2 - 10 Matrix multiplication: The i, j entry of the matrix product is the dot product of row i of the left matrix with column j of the right one.

11. 2 - 11 Matrix multiplication: Notes: Notes: (1) A+B = B+A, (2)

12. 2 - 12 Matrix form of a system of linear equations: === A x b

13. 2 - 13 a linear combination of the column vectors of matrix A: linear combination of column vectors of A ===

14. 2 - 14 2A.2 Properties of Matrix Operations Three basic matrix operators: (1) matrix addition (2) scalar multiplication (3) matrix multiplication Zero matrix: Identity matrix of order n:

15. 2 - 15 Then (1) A+B = B + A (2) A + ( B + C ) = ( A + B ) + C (3) ( cd ) A = c ( dA ) (4) 1A = A (5) c( A+B ) = cA + cB (6) ( c+d ) A = cA + dA Properties of matrix addition and scalar multiplication: Commutative law for addition Associative law for addition Associative law for scalar multiplication Unit element for scalar multiplication Distributive law 1 for scalar multiplication Distributive law 2 for scalar multiplication

16. 2 - 16 Properties of zero matrices: the additive identity for all m×n matrices the additive inverse of A

17. 2 - 17 Properties of the multiplicative identity matrix: Properties of matrix multiplication:

18. 2 - 18 Transpose of a matrix: Properties of transposes:

19. 2 - 19 A square matrix A is symmetric if A = A T A square matrix A is skew-symmetric if A T = –A Skew-symmetric matrix: Symmetric matrix: Note: is symmetric Pf:

20. 2 - 20 ab = ba(Commutative law for multiplication) (Sizes are not the same) (Sizes are the same, but matrices are not equal) Real number: Matrix: Three situations of the non-commutativity may occur :

21. 2 - 21 Sol: Example (Case 3): Show that AB and BA are not equal for the matrices. and

22. 2 - 22 (Cancellation is not valid) (Cancellation law) Matrix: (1) If C is invertible (i.e., C -1 exists), then A = B Real number:

23. 2 - 23 Sol: So But Example: An example in which cancellation is not valid Show that AC=BC C is noninvertible, (i.e., row 1 and row 2 are not independent)

24. 2 - 24 2A.3 The Inverse of a Matrix Note: A matrix that does not have an inverse is called noninvertible (or singular). Consider Then (1) A is invertible (or nonsingular) (2) B is the inverse of A Inverse matrix:

25. 2 - 25 If B and C are both inverses of the matrix A, then B = C. Pf: the inverse of a matrix is unique. Consequently, the inverse of a matrix is unique. Notes: (1) The inverse of A is denoted by Theorem: Theorem: The inverse of a matrix is unique

26. 2 - 26 Example: Find the inverse of the matrix Sol: A Find the inverse of a matrix A by Gauss-Jordan Elimination:

27. 2 - 27 Thus

28. 2 - 28 If A can’t be row reduced to I, then A is singular. Note:

29. 2 - 29 Sol: Example: Find the inverse of the following matrix

30. 2 - 30 So the matrix A is invertible, and its inverse is You can check:

31. 2 - 31 square matrix Power of a square matrix:

32. 2 - 32 If A is an invertible matrix, k is a positive integer, and c is a scalar, then Theorem ： Theorem ： Properties of inverse matrices

33. 2 - 33 Theorem: Theorem: The inverse of a product If A and B are invertible matrices of size n, then AB is invertible and Pf: Note:

34. 2 - 34 Theorem: Theorem: Cancellation properties If C is an invertible matrix, then the following properties hold: (1) If AC=BC, then A=B (Right cancellation property) (2) If CA=CB, then A=B (Left cancellation property) Pf: Note: If C is not invertible, then cancellation is not valid.

35. 2 - 35 Theorem: Theorem: Systems of linear equations with unique solutions If A is an invertible matrix, then the system of linear equations Ax = b has a unique solution given by Pf: ( A is nonsingular) This solution is unique. (Left cancellation property)

36. 2 - 36 Note: Only do a single elementary row operation. 2A.4 Elementary Matrices Row elementary matrix: elementary matrix An n  n matrix is called an elementary matrix if it can be obtained from the identity matrix I by a single elementary row operation.

37. 2 - 37 Ex: (Elementary matrices and nonelementary matrices)

38. 2 - 38 Notes: Theorem: Theorem: Representing elementary row operations Let E be the elementary matrix obtained by performing an elementary row operation on I m. If that same elementary row operation is performed on an m  n matrix A, then the resulting matrix is given by the product EA.

39. 2 - 39 Sol: Example: Using elementary matrices Find a sequence of elementary matrices that can be used to write the matrix A in row-echelon form.

40. 2 - 40 row-echelon form

41. 2 - 41 Matrix B is row-equivalent to A if there exists a finite number of elementary matrices such that Row-equivalent:

42. 2 - 42 Theorem : Theorem : Elementary matrices are invertible If E is an elementary matrix, then exists and is an elementary matrix. Notes:

43. 2 - 43 Ex: Elementary Matrix Inverse Matrix

44. 2 - 44 Pf: (1)Assume that A is the product of elementary matrices. (a) Every elementary matrix is invertible. (b) The product of invertible matrices is invertible. Thus A is invertible. (2) If A is invertible, has only the trivial solution. Thus A can be written as the product of elementary matrices. Theorem: Theorem: A property of invertible matrices A square matrix A is invertible if and only if it can be written as the product of elementary matrices.

45. 2 - 45 Sol: Ex: Find a sequence of elementary matrices whose product is

46. 2 - 46 Note: If A is invertible

47. 2 - 47 If A is an n  n matrix, then the following statements are equivalent. (1) A is invertible. (2) Ax = b has a unique solution for every n  1 column matrix b. (3) Ax = 0 has only the trivial solution. (4) A is row-equivalent to I n. (5) A can be written as the product of elementary matrices. Thm: Thm: Equivalent conditions

48. 2 - 48 L is a lower triangular matrix U is an upper triangular matrix lower A n  n matrix A can be written as the product of a lower triangular matrixLupper triangular matrixU triangular matrix L and an upper triangular matrix U, such as Note: If a square matrix A can be row reduced to an upper triangular only the row operation of adding a multiple of one matrix U using only the row operation of adding a multiple of one row to another (pivot) row to another (pivot), then it is easy to find an LU-factorization of A. LU-factorization:

49. 2 - 49 Sol: (a) Ex: LU-factorization

50. 2 - 50 (b)

51. 2 - 51 Two steps: (1) Write y = Ux and solve Ly = b for y (2) Solve Ux = y for x Ax=b Solving Ax=b with an LU-factorization of A

52. 2 - 52 Sol: Ex: Solving a linear system using LU-factorization

53. 2 - 53 Thus, the solution is So