1. Matrices & Systems of Linear Equations

3. Special Matrices

4. Special Matrices

5. corresponding entries are equal
Equality of Matrices
Two matrices are said
to be equal if they have
the same size and their
corresponding entries
are equal

6. Equality of Matrices
Use the given equality
to find x, y and z

7. Matrix Addition and Subtraction Example (1)

8. Matrix Addition and Subtraction Example (2)

9. Multiplication of a Matrix by a Scalar

10. The result is a (n by k) Matrix
Matrix Multiplication (n by m) Matrix X (m by k) Matrix The number of columns of the matrix on the left = number of rows of the matrix on the right
The result is a (n by k) Matrix

11. Matrix Multiplication 3x3 X 3x3

12. Matrix Multiplication 1x3 X 3x3→ 1x3

13. Example (1)

14. Example (2) (1X3) X (3X3) → 1X3

15. Example (3) (3X1) X (1X2) → 3X2

16. Example (4)

17. Transpose of Matrix

20. Properties of the Transpose

21. Matrix Reduction Definitions (1)
1. Zero Row: A row consisting entirely of zeros
2. Nonzero Row: A row having at least one nonzero entry
3. Leading Entry of a row: The first nonzero entry of a row.

22. Matrix Reduction Definitions (2)
Reduced Matrix: A matrix satisfying the following:
1. All zero rows, if any, are at the bottom of the matrix
2. The leading entry of a row is 1
3. All other entries in the column in which the leading entry is located are zeros.
4. A leading entry in a row is to the right of a leading entry in any row above it.

23. Examples of Reduced Matrices

24. Examples matrices that are not reduced

26. Elementary Row Operations
1. Interchanging two rows
2. Replacing a row by a nonzero multiple of itself
3. Replacing a row by the sum of that row and a nonzero multiple of another row.

27. Interchanging Rows

28. Replacing a row by a nonzero multiple of itself

29. Replacing a row by the sum of that row and a nonzero multiple of another row

30. Augmented Matrix Representing a System of linear Equations

32. Solving a System of Linear Equations by Reducing its Augmented Matrix Using Row Operations

35. Solution

37. Solution of the System

38. The Idea behind the Reduction Method

40. Interchanging the First & the Second Row

41. Multiplying the first Equation by 1/3

42. Subtracting from the Third Equation 5 times the First Equation

43. Subtracting from the First Equation 2 times the Second Equation

44. Adding to the Third Equation 12 times the Second Equation

45. Dividing the Third Equation by 40

46. Adding to the First Equation 7 times the third Equation

47. Subtracting from the Second Equation 3/2 times the third Equation

48. Systems with infinitely many Solutions
x=3-2r
y = r 3 5 -1 1
-17 10

50. Systems with infinitely many Solutions
y=-r
x=-3r
z=r -1 -3 1 10 30
-10
-1/3
1/3

52. Details of reduction

53. Systems with no Solution

57. Details of the reduction

59. Finding the Inverse of an nXn square Matrix A
1. Adjoin the In identity matrix to obtain the Augmented matrix [A| In ]
2. Reduce [A| In ] to [In | B ] if possible
Then
B = A-1

60. Example (1)

62. Example (2)

64. Inverse Matrix The formula for the inverse of a 2X2 Matrix

65. Using the Inverse Matrix to Solve System of Linear Equations

66. Problem

68. Homework