Systems and Matrices

1
(No Transcript)
2
Chapter 7

• Systems and Matrices

3
7.1

• Solving Systems of Two Equations

4
Quick Review

5
What youll learn about

• The Method of Substitution
• Solving Systems Graphically
• The Method of Elimination
• Applications
• and why
• Many applications in business and science can be
• modeled using systems of equations.

6
Solution of a System

• A solution of a system of two equations in two
• variables is an ordered pair of real numbers that
  
• is a solution of each equation.

7
Example Using the Substitution Method

8
Example Using the Substitution Method

9
Example Solving a Nonlinear System Algebraically

10
Example Using the Elimination Method

11
Example Finding No Solution

12
Example Finding Infinitely Many Solutions

13
7.2

• Matrix Algebra

14
Quick Review

15
What youll learn about

• Matrices
• Matrix Addition and Subtraction
• Matrix Multiplication
• Identity and Inverse Matrices
• Determinant of a Square Matrix
• Applications
• and why
• Matrix algebra provides a powerful technique to
  manipulate large
• data sets and solve the related problems that are
  modeled by the
• matrices.

16
Matrix

17
Matrix Vocabulary

• Each element, or entry, aij, of the matrix uses
• double subscript notation. The row subscript is
• the first subscript i, and the column subscript
  is
• j. The element aij is the ith row and the jth
• column. In general, the order of an m n
• matrix is mn.

18
Example Determining the Order of a Matrix

19
Matrix Addition and Matrix Subtraction

20
Adding Matrices
21
Example Matrix Addition

22
Example Using Scalar Multiplication

23
The Zero Matrix

24
Additive Inverse

25
Matrix Multiplication

26
Multiplying Matrices
Let A denote an m by r matrix and let B denote an
r by n matrix. The product AB is defined as the
m by n matrix whose entry in row i, column j is
the product of the ith row of A and the jth
column of B.
Note If we multiply a matrix by a constant,
this is equivalent to multiplying each term in
the matrix by the constant.
27
Example Matrix Multiplication

28
Example Matrix Multiplication

29
Example Find the product AB
30
The Identity Matrix
31
Identity Matrix

32
Inverse of a Square Matrix

33
Inverse of a 2 2 Matrix

34
Determinant of a Square Matrix

35
Inverses of n n Matrices

• An n n matrix A has an inverse if and only if
• det A ? 0.

36
Example Finding Inverse Matrices

37
Properties of Matrices

• Let A, B, and C be matrices whose orders are such
  that the following sums, differences, and
  products are defined.
• 1. Community property
• Addition A B B A
• Multiplication Does not hold in general
• 2. Associative property
• Addition (A B) C A (B C)
• Multiplication (AB)C A(BC)
• 3. Identity property
• Addition A 0 A
• Multiplication AIn InA A
• 4. Inverse property
• Addition A (-A) 0
• Multiplication AA-1 A-1A In A?0
• 5. Distributive property
• Multiplication over addition A(B C) AB AC
  (A B)C AC BC
• Multiplication over subtraction A(B - C) AB -
  AC (A - B)C AC - BC

38
Matrices and Transformations
39
Matrices and Transformations
Rotation through an angle ? The rotation through
an angle ? maps each point P(x,y) in the
rectangular coordinate plane to the point
P(x,y). where
40
Matrices and Transformations
or
41
Matrices and Transformations
Find the rotation matrix about the origin whose
angle is ?/3.
42
Matrices and Transformations
Where does the point (4,-2) move?
43
7.3

• Multivariate Linear Systems and Row Operations

44
Quick Review

45
What youll learn about

• Triangular Forms for Linear Systems
• Gaussian Elimination
• Elementary Row Operations and Row Echelon Form
• Reduced Row Echelon Form
• Solving Systems with Inverse Matrices
• Applications
• and why
• Many applications in business and science are
  modeled by
• systems of linear equations in three or more
  variables.

46
Equivalent Systems of Linear Equations

• The following operations produce an equivalent
• system of linear equations.
• Interchange any two equations of the system.
• Multiply (or divide) one of the equations by any
  nonzero real number.
• Add a multiple of one equation to any other
  equation in the system.

47
Row Echelon Form of a Matrix

• A matrix is in row echelon form if the following
• conditions are satisfied.
• Rows consisting entirely of 0s (if there are
  any) occur at the bottom of the matrix.
• The first entry in any row with nonzero entries
  is 1.
• The column subscript of the leading 1 entries
  increases as the row subscript increases.

48
Elementary Row Operations on a Matrix

• A combination of the following operations will
• transform a matrix to row echelon form.
• Interchange any two rows.
• Multiply all elements of a row by a nonzero real
  number.
• Add a multiple of one row to any other row.

49
Example Finding a Row Echelon Form

50
Example Finding a Row Echelon Form

51
Reduced Row Echelon Form

• If we continue to apply elementary row
• operations to a row echelon form of a matrix, we
• can obtain a matrix in which every column that
• has a leading 1 has 0s elsewhere. This is the
• reduced echelon form.

52
Example Solving a System Using Inverse Matrices

53
Example Solving a System Using Inverse Matrices

54
Multivariate Linear Systems and Row Operations
Page 602
55
Multivariate Linear Systems and Row Operations
Page 602
56
Multivariate Linear Systems and Row Operations
57
7.4

• Partial Fractions

58
Quick Review

59
What youll learn about

• Partial Fraction Decomposition
• Denominators with Linear Factors
• Denominators with Irreducible Quadratic Factors
• Applications
• and why
• Partial fraction decompositions are used in
  calculus in
• integration and can be used to guide the sketch
  of the
• graph of a rational function.

60
Partial Fraction Decomposition of f(x)/d(x)

61
Example Decomposing a Fraction with Distinct
Linear Factors

62
Example Decomposing a Fraction with Distinct
Linear Factors

63
Example Decomposing a Fraction with an
Irreducible Quadratic Factor

64
Example Decomposing a Fraction with an
Irreducible Quadratic Factor

65
7.5

• Systems of Inequalities in Two Variables

66
Quick Review Solutions

67
What youll learn about

• Graph of an Inequality
• Systems of Inequalities
• Linear Programming
• and why
• Linear programming is used in business and
• industry to maximize profits, minimize costs, and
  to
• help management make decisions.

68
Steps for Drawing the Graph of an Inequality in
Two Variables

1. Draw the graph of the equation obtained by
   replacing the inequality sign by an equal sign.
   Use a dashed line if the inequality is lt
   orgt. Use a solid line if the inequality is
   or .
2. Check a point in each of the two regions of the
   plane determined by the graph of the equation. If
   the point satisfies the inequality, then shade
   the region containing the point.

69
Example Graphing a Linear Inequality

70
Example Graphing a Linear Inequality

71
Example Solving a System of Inequalities
Graphically

72
Example Solving a System of Inequalities
Graphically

73
Chapter Test

74
Chapter Test

75
Chapter Test

76
Chapter Test

77
Chapter Test Solutions

78
Chapter Test Solutions

79
Chapter Test Solutions

80
Chapter Test Solutions