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INSTRUCTORS Carleen Eaton Grant Fraser
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Dr. Carleen Eaton

Dr. Carleen Eaton

Identity and Inverse Matrices

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Table of Contents

I. Equations and Inequalities
Expressions and Formulas

22m 23s

Intro
0:00
Order of Operations
0:19
Variable
0:27
Algebraic Expression
0:46
Term
0:57
Example: Algebraic Expression
1:25
Evaluate Inside Grouping Symbols
1:55
Evaluate Powers
2:30
Multiply/Divide Left to Right
2:55
Add/Subtract Left to Right
3:35
Monomials
4:40
Examples of Monomials
4:52
Constant
5:27
Coefficient
5:46
Degree
6:25
Power
7:15
Polynomials
8:02
Examples of Polynomials
8:24
Binomials, Trinomials, Monomials
8:53
Term
9:21
Like Terms
10:02
Formulas
11:00
Example: Pythagorean Theorem
11:15
Example 1: Evaluate the Algebraic Expression
11:50
Example 2: Evaluate the Algebraic Expression
14:38
Example 3: Area of a Triangle
19:11
Example 4: Fahrenheit to Celsius
20:41
Properties of Real Numbers

20m 15s

Intro
0:00
Real Numbers
0:07
Number Line
0:15
Rational Numbers
0:46
Irrational Numbers
2:24
Venn Diagram of Real Numbers
4:03
Irrational Numbers
5:00
Rational Numbers
5:19
Real Number System
5:27
Natural Numbers
5:32
Whole Numbers
5:53
Integers
6:19
Fractions
6:46
Properties of Real Numbers
7:15
Commutative Property
7:34
Associative Property
8:07
Identity Property
9:04
Inverse Property
9:53
Distributive Property
11:03
Example 1: What Set of Numbers?
12:21
Example 2: What Properties Are Used?
13:56
Example 3: Multiplicative Inverse
16:00
Example 4: Simplify Using Properties
17:18
Solving Equations

19m 10s

Intro
0:00
Translations
0:06
Verbal Expressions and Algebraic Expressions
0:13
Example: Sum of Two Numbers
0:19
Example: Square of a Number
1:33
Properties of Equality
3:20
Reflexive Property
3:30
Symmetric Property
3:42
Transitive Property
4:01
Addition Property
5:01
Subtraction Property
5:37
Multiplication Property
6:02
Division Property
6:30
Solving Equations
6:58
Example: Using Properties
7:18
Solving for a Variable
8:25
Example: Solve for Z
8:34
Example 1: Write Algebraic Expression
10:15
Example 2: Write Verbal Expression
11:31
Example 3: Solve the Equation
14:05
Example 4: Simplify Using Properties
17:26
Solving Absolute Value Equations

17m 31s

Intro
0:00
Absolute Value Expressions
0:09
Distance from Zero
0:18
Example: Absolute Value Expression
0:24
Absolute Value Equations
1:50
Example: Absolute Value Equation
2:00
Example: Isolate Expression
3:13
No Solution
3:46
Empty Set
3:58
Example: No Solution
4:12
Number of Solutions
4:46
Check Each Solution
4:57
Example: Two Solutions
5:05
Example: No Solution
6:18
Example: One Solution
6:28
Example 1: Evaluate for X
7:16
Example 2: Write Verbal Expression
9:08
Example 3: Solve the Equation
12:18
Example 4: Simplify Using Properties
13:36
Solving Inequalities

17m 14s

Intro
0:00
Properties of Inequalities
0:08
Addition Property
0:17
Example: Using Numbers
0:30
Subtraction Property
1:03
Example: Using Numbers
1:19
Multiplication Properties
1:44
C>0 (Positive Number)
1:50
Example: Using Numbers
2:05
C<0 (Negative Number)
2:40
Example: Using Numbers
3:10
Division Properties
4:11
C>0 (Positive Number)
4:15
Example: Using Numbers
4:27
C<0 (Negative Number)
5:21
Example: Using Numbers
5:32
Describing the Solution Set
6:10
Example: Set Builder Notation
6:26
Example: Graph (Closed Circle)
7:08
Example: Graph (Open Circle)
7:30
Example 1: Solve the Inequality
7:58
Example 2: Solve the Inequality
9:06
Example 3: Solve the Inequality
10:10
Example 4: Solve the Inequality
13:12
Solving Compound and Absolute Value Inequalities

25m

Intro
0:00
Compound Inequalities
0:08
And and Or
0:13
Example: And
0:22
Example: Or
1:12
And Inequality
1:41
Intersection
1:49
Example: Numbers
2:08
Example: Inequality
2:43
Or Inequality
4:35
Example: Union
4:45
Example: Inequality
5:53
Absolute Value Inequalities
7:19
Definition of Absolute Value
7:33
Examples: Compound Inequalities
8:30
Example: Complex Inequality
12:21
Example 1: Solve the Inequality
12:54
Example 2: Solve the Inequality
17:21
Example 3: Solve the Inequality
18:54
Example 4: Solve the Inequality
22:15
II. Linear Relations and Functions
Relations and Functions

32m 5s

Intro
0:00
Coordinate Plane
0:20
X-Coordinate and Y-Coordinate
0:30
Example: Coordinate Pairs
0:37
Quadrants
1:20
Relations
2:14
Domain and Range
2:19
Set of Ordered Pairs
2:29
As a Table
2:51
Functions
4:21
One Element in Range
4:32
Example: Mapping
4:43
Example: Table and Map
6:26
One-to-One Functions
8:01
Example: One-to-One
8:22
Example: Not One-to-One
9:18
Graphs of Relations
11:01
Discrete and Continuous
11:12
Example: Discrete
11:22
Example: Continous
12:30
Vertical Line Test
14:09
Example: S Curve
14:29
Example: Function
16:15
Equations, Relations, and Functions
17:03
Independent Variable and Dependent Variable
17:16
Function Notation
19:11
Example: Function Notation
19:23
Example 1: Domain and Range
20:51
Example 2: Discrete or Continous
23:03
Example 3: Discrete or Continous
25:53
Example 4: Function Notation
30:05
Linear Equations

14m 46s

Intro
0:00
Linear Equations and Functions
0:07
Linear Equation
0:19
Example: Linear Equation
0:29
Example: Linear Function
1:07
Standard Form
2:02
Integer Constants with No Common Factor
2:08
Example: Standard Form
2:27
Graphing with Intercepts
4:05
X-Intercept and Y-Intercept
4:12
Example: Intercepts
4:26
Example: Graphing
5:14
Example 1: Linear Function
7:53
Example 2: Linear Function
9:10
Example 3: Standard Form
10:04
Example 4: Graph with Intercepts
12:25
Slope

23m 7s

Intro
0:00
Definition of Slope
0:07
Change in Y / Change in X
0:26
Example: Slope of Graph
0:37
Interpretation of Slope
3:07
Horizontal Line (0 Slope)
3:13
Vertical Line (Undefined Slope)
4:52
Rises to Right (Positive Slope)
6:36
Falls to Right (Negative Slope)
6:53
Parallel Lines
7:18
Example: Not Vertical
7:30
Example: Vertical
7:58
Perpendicular Lines
8:31
Example: Perpendicular
8:42
Example 1: Slope of Line
10:32
Example 2: Graph Line
11:45
Example 3: Parallel to Graph
13:37
Example 4: Perpendicular to Graph
17:57
Writing Linear Functions

23m 5s

Intro
0:00
Slope Intercept Form
0:11
m and b
0:28
Example: Graph Using Slope Intercept
0:43
Point Slope Form
2:41
Relation to Slope Formula
3:03
Example: Point Slope Form
4:36
Parallel and Perpendicular Lines
6:28
Review of Parallel and Perpendicular Lines
6:31
Example: Parallel
7:50
Example: Perpendicular
9:58
Example 1: Slope Intercept Form
11:07
Example 2: Slope Intercept Form
13:07
Example 3: Parallel
15:49
Example 4: Perpendicular
18:42
Special Functions

31m 5s

Intro
0:00
Step Functions
0:07
Example: Apple Prices
0:30
Absolute Value Function
4:55
Example: Absolute Value
5:05
Piecewise Functions
9:08
Example: Piecewise
9:27
Example 1: Absolute Value Function
14:00
Example 2: Absolute Value Function
20:39
Example 3: Piecewise Function
22:26
Example 4: Step Function
25:25
Graphing Inequalities

21m 42s

Intro
0:00
Graphing Linear Inequalities
0:07
Shaded Region
0:19
Using Test Points
0:32
Graph Corresponding Linear Function
0:46
Dashed or Solid Lines
0:59
Use Test Point
1:21
Example: Linear Inequality
1:58
Graphing Absolute Value Inequalities
4:50
Graph Corresponding Equations
4:59
Use Test Point
5:20
Example: Absolute Value Inequality
5:38
Example 1: Linear Inequality
9:17
Example 2: Linear Inequality
11:56
Example 3: Linear Inequality
14:29
Example 4: Absolute Value Inequality
17:06
III. Systems of Equations and Inequalities
Solving Systems of Equations by Graphing

17m 13s

Intro
0:00
Systems of Equations
0:09
Example: Two Equations
0:24
Solving by Graphing
0:53
Point of Intersection
1:09
Types of Systems
2:29
Independent (Single Solution)
2:34
Dependent (Infinite Solutions)
3:05
Inconsistent (No Solution)
4:23
Example 1: Solve by Graphing
5:20
Example 2: Solve by Graphing
9:10
Example 3: Solve by Graphing
12:27
Example 4: Solve by Graphing
14:54
Solving Systems of Equations Algebraically

23m 53s

Intro
0:00
Solving by Substitution
0:08
Example: System of Equations
0:36
Solving by Multiplication
7:22
Extra Step of Multiplying
7:38
Example: System of Equations
8:00
Inconsistent and Dependent Systems
11:14
Variables Drop Out
11:48
Inconsistent System (Never True)
12:01
Constant Equals Constant
12:53
Dependent System (Always True)
13:11
Example 1: Solve Algebraically
13:58
Example 2: Solve Algebraically
15:52
Example 3: Solve Algebraically
17:54
Example 4: Solve Algebraically
21:40
Solving Systems of Inequalities By Graphing

27m 12s

Intro
0:00
Solving by Graphing
0:08
Graph Each Inequality
0:25
Overlap
0:35
Corresponding Linear Equations
1:03
Test Point
1:23
Example: System of Inequalities
1:51
No Solution
7:06
Empty Set
7:26
Example: No Solution
7:34
Example 1: Solve by Graphing
10:27
Example 2: Solve by Graphing
13:30
Example 3: Solve by Graphing
17:19
Example 4: Solve by Graphing
23:23
Solving Systems of Equations in Three Variables

28m 53s

Intro
0:00
Solving Systems in Three Variables
0:17
Triple of Values
0:31
Example: Three Variables
0:56
Number of Solutions
5:55
One Solution
6:08
No Solution
6:24
Infinite Solutions
7:06
Example 1: Solve 3 Variables
7:59
Example 2: Solve 3 Variables
13:50
Example 3: Solve 3 Variables
19:54
Example 4: Solve 3 Variables
25:50
IV. Matrices
Basic Matrix Concepts

11m 34s

Intro
0:00
What is a Matrix
0:26
Brackets
0:46
Designation
1:21
Element
1:47
Matrix Equations
1:59
Dimensions
2:27
Rows (m) and Columns (n)
2:37
Examples: Dimensions
2:43
Special Matrices
4:22
Row Matrix
4:32
Column Matrix
5:00
Zero Matrix
6:00
Equal Matrices
6:30
Example: Corresponding Elements
6:36
Example 1: Matrix Dimension
8:12
Example 2: Matrix Dimension
9:03
Example 3: Zero Matrix
9:38
Example 4: Row and Column Matrix
10:26
Matrix Operations

21m 36s

Intro
0:00
Matrix Addition
0:18
Same Dimensions
0:25
Example: Adding Matrices
1:04
Matrix Subtraction
3:42
Same Dimensions
3:48
Example: Subtracting Matrices
4:04
Scalar Multiplication
6:08
Scalar Constant
6:24
Example: Multiplying Matrices
6:32
Properties of Matrix Operations
8:23
Commutative Property
8:41
Associative Property
9:08
Distributive Property
9:44
Example 1: Matrix Addition
10:24
Example 2: Matrix Subtraction
11:58
Example 3: Scalar Multiplication
14:23
Example 4: Matrix Properties
16:09
Matrix Multiplication

29m 36s

Intro
0:00
Dimension Requirement
0:17
n = p
0:24
Resulting Product Matrix (m x q)
1:21
Example: Multiplication
1:54
Matrix Multiplication
3:38
Example: Matrix Multiplication
4:07
Properties of Matrix Multiplication
10:46
Associative Property
11:00
Associative Property (Scalar)
11:28
Distributive Property
12:06
Distributive Property (Scalar)
12:30
Example 1: Possible Matrices
13:31
Example 2: Multiplying Matrices
17:08
Example 3: Multiplying Matrices
20:41
Example 4: Matrix Properties
24:41
Determinants

33m 13s

Intro
0:00
What is a Determinant
0:13
Square Matrices
0:23
Vertical Bars
0:41
Determinant of a 2x2 Matrix
1:21
Second Order Determinant
1:37
Formula
1:45
Example: 2x2 Determinant
1:58
Determinant of a 3x3 Matrix
2:50
Expansion by Minors
3:08
Third Order Determinant
3:19
Expanding Row One
4:06
Example: 3x3 Determinant
6:40
Diagonal Method for 3x3 Matrices
13:24
Example: Diagonal Method
13:36
Example 1: Determinant of 2x2
18:59
Example 2: Determinant of 3x3
20:03
Example 3: Determinant of 3x3
25:35
Example 4: Determinant of 3x3
29:22
Cramer's Rule

28m 25s

Intro
0:00
System of Two Equations in Two Variables
0:16
One Variable
0:50
Determinant of Denominator
1:14
Determinants of Numerators
2:23
Example: System of Equations
3:34
System of Three Equations in Three Variables
7:06
Determinant of Denominator
7:17
Determinants of Numerators
7:52
Example 1: Two Equations
8:57
Example 2: Two Equations
13:21
Example 3: Three Equations
17:11
Example 4: Three Equations
23:43
Identity and Inverse Matrices

22m 25s

Intro
0:00
Identity Matrix
0:13
Example: 2x2 Identity Matrix
0:30
Example: 4x4 Identity Matrix
0:50
Properties of Identity Matrices
1:24
Example: Multiplying Identity Matrix
2:52
Matrix Inverses
5:30
Writing Matrix Inverse
6:07
Inverse of a 2x2 Matrix
6:39
Example: 2x2 Matrix
7:31
Example 1: Inverse Matrix
10:18
Example 2: Find the Inverse Matrix
13:04
Example 3: Find the Inverse Matrix
17:53
Example 4: Find the Inverse Matrix
20:44
Solving Systems of Equations Using Matrices

22m 32s

Intro
0:00
Matrix Equations
0:11
Example: System of Equations
0:21
Solving Systems of Equations
4:01
Isolate x
4:16
Example: Using Numbers
5:10
Multiplicative Inverse
5:54
Example 1: Write as Matrix Equation
7:18
Example 2: Use Matrix Equations
9:12
Example 3: Use Matrix Equations
15:06
Example 4: Use Matrix Equations
19:35
V. Quadratic Functions and Inequalities
Graphing Quadratic Functions

31m 48s

Intro
0:00
Quadratic Functions
0:12
A is Zero
0:27
Example: Parabola
0:45
Properties of Parabolas
2:08
Axis of Symmetry
2:11
Vertex
2:32
Example: Parabola
2:48
Minimum and Maximum Values
9:02
Positive or Negative
9:28
Upward or Downward
9:58
Example: Minimum
10:31
Example: Maximum
11:16
Example 1: Axis of Symmetry, Vertex, Graph
12:41
Example 2: Axis of Symmetry, Vertex, Graph
17:25
Example 3: Minimum or Maximum
21:47
Example 4: Minimum or Maximum
27:09
Solving Quadratic Equations by Graphing

27m 3s

Intro
0:00
Quadratic Equations
0:16
Standard Form
0:18
Example: Quadratic Equation
0:47
Solving by Graphing
1:41
Roots (x-Intercepts)
1:48
Example: Number of Solutions
2:12
Estimating Solutions
9:23
Example: Integer Solutions
9:30
Example: Estimating
9:53
Example 1: Solve by Graphing
10:52
Example 2: Solve by Graphing
15:10
Example 1: Solve by Graphing
17:50
Example 1: Solve by Graphing
20:54
Solving Quadratic Equations by Factoring

19m 53s

Intro
0:00
Factoring Techniques
0:15
Greatest Common Factor (GCF)
0:37
Difference of Two Squares
1:48
Perfect Square Trinomials
2:30
General Trinomials
3:09
Zero Product Rule
5:22
Example: Zero Product
5:53
Example 1: Solve by Factoring
7:46
Example 1: Solve by Factoring
9:48
Example 1: Solve by Factoring
12:34
Example 1: Solve by Factoring
15:28
Imaginary and Complex Numbers

35m 45s

Intro
0:00
Properties of Square Roots
0:10
Product Property
0:26
Example: Product Property
0:56
Quotient Property
2:17
Example: Quotient Property
2:35
Imaginary Numbers
3:12
Imaginary i
3:51
Examples: Imaginary Number
4:22
Complex Numbers
7:23
Real Part and Imaginary Part
7:33
Examples: Complex Numbers
7:57
Equality
9:37
Example: Equal Complex Numbers
9:52
Addition and Subtraction
10:12
Examples: Adding Complex Numbers
10:25
Complex Plane
13:32
Horizontal Axis (Real)
13:49
Vertical Axis (Imaginary)
13:59
Example: Labeling
14:11
Multiplication
15:57
Example: FOIL Method
16:03
Division
18:37
Complex Conjugates
18:45
Conjugate Pairs
19:10
Example: Dividing Complex Numbers
20:00
Example 1: Simplify Complex Number
24:50
Example 2: Simplify Complex Number
27:56
Example 3: Multiply Complex Numbers
29:27
Example 3: Dividing Complex Numbers
31:48
Completing the Square

27m 11s

Intro
0:00
Square Root Property
0:12
Example: Perfect Square
0:38
Example: Perfect Square Trinomial
3:00
Completing the Square
4:39
Constant Term
4:50
Example: Complete the Square
5:04
Solve Equations
6:42
Add to Both Sides
6:59
Example: Complete the Square
7:07
Equations Where a Not Equal to 1
10:58
Divide by Coefficient
11:08
Example: Complete the Square
11:24
Complex Solutions
14:05
Real and Imaginary
14:14
Example: Complex Solution
14:35
Example 1: Square Root Property
18:31
Example 2: Complete the Square
19:15
Example 3: Complete the Square
20:40
Example 4: Complete the Square
23:56
Quadratic Formula and the Discriminant

22m 48s

Intro
0:00
Quadratic Formula
0:21
Standard Form
0:29
Example: Quadratic Formula
0:57
One Rational Root
3:00
Example: One Root
3:31
Complex Solutions
6:16
Complex Conjugate
6:28
Example: Complex Solution
7:15
Discriminant
9:42
Positive Discriminant
10:03
Perfect Square (Rational)
10:51
Not Perfect Square (2 Irrational)
11:27
Negative Discriminant
12:28
Zero Discriminant
12:57
Example 1: Quadratic Formula
13:50
Example 2: Quadratic Formula
16:03
Example 3: Quadratic Formula
19:00
Example 4: Discriminant
21:33
Analyzing the Graphs of Quadratic Functions

30m 7s

Intro
0:00
Vertex Form
0:12
H and K
0:32
Axis of Symmetry
0:36
Vertex
0:42
Example: Origin
1:00
Example: k = 2
2:12
Example: h = 1
4:27
Significance of Coefficient a
7:13
Example: |a| > 1
7:25
Example: |a| < 1
8:18
Example: |a| > 0
8:51
Example: |a| < 0
9:05
Writing Quadratic Equations in Vertex Form
10:22
Standard Form to Vertex Form
10:35
Example: Standard Form
11:02
Example: a Term Not 1
14:42
Example 1: Vertex Form
19:47
Example 2: Vertex Form
22:09
Example 3: Vertex Form
24:32
Example 4: Vertex Form
28:23
Graphing and Solving Quadratic Inequalities

27m 5s

Intro
0:00
Graphing Quadratic Inequalities
0:11
Test Point
0:18
Example: Quadratic Inequality
0:29
Solving Quadratic Inequalities
3:57
Example: Parameter
4:24
Example 1: Graph Inequality
11:16
Example 2: Solve Inequality
14:27
Example 3: Graph Inequality
19:14
Example 4: Solve Inequality
23:48
VI. Polynomial Functions
Properties of Exponents

19m 29s

Intro
0:00
Simplifying Exponential Expressions
0:09
Monomial Simplest Form
0:19
Negative Exponents
1:07
Examples: Simple
1:34
Properties of Exponents
3:06
Negative Exponents
3:13
Mutliplying Same Base
3:24
Dividing Same Base
3:45
Raising Power to a Power
4:33
Parentheses (Multiplying)
5:11
Parentheses (Dividing)
5:47
Raising to 0th Power
6:15
Example 1: Simplify Exponents
7:59
Example 2: Simplify Exponents
10:41
Example 3: Simplify Exponents
14:11
Example 4: Simplify Exponents
18:04
Operations on Polynomials

13m 27s

Intro
0:00
Adding and Subtracting Polynomials
0:13
Like Terms and Like Monomials
0:23
Examples: Adding Monomials
1:14
Multiplying Polynomials
3:40
Distributive Property
3:44
Example: Monomial by Polynomial
4:06
Example 1: Simplify Polynomials
5:47
Example 2: Simplify Polynomials
6:28
Example 3: Simplify Polynomials
8:38
Example 4: Simplify Polynomials
10:47
Dividing Polynomials

31m 11s

Intro
0:00
Dividing by a Monomial
0:13
Example: Numbers
0:26
Example: Polynomial by a Monomial
1:18
Long Division
2:28
Remainder Term
2:41
Example: Dividing with Numbers
3:04
Example: With Polynomials
5:01
Example: Missing Terms
7:58
Synthetic Division
11:44
Restriction
12:04
Example: Divisor in Form
12:20
Divisor in Synthetic Division
15:54
Example: Coefficient to 1
16:07
Example 1: Divide Polynomials
17:10
Example 2: Divide Polynomials
19:08
Example 3: Synthetic Division
21:42
Example 4: Synthetic Division
25:09
Polynomial Functions

22m 30s

Intro
0:00
Polynomial in One Variable
0:13
Leading Coefficient
0:27
Example: Polynomial
1:18
Degree
1:31
Polynomial Functions
2:57
Example: Function
3:13
Function Values
3:33
Example: Numerical Values
3:53
Example: Algebraic Expressions
5:11
Zeros of Polynomial Functions
5:50
Odd Degree
6:04
Even Degree
7:29
End Behavior
8:28
Even Degrees
9:09
Example: Leading Coefficient +/-
9:23
Odd Degrees
12:51
Example: Leading Coefficient +/-
13:00
Example 1: Degree and Leading Coefficient
15:03
Example 2: Polynomial Function
15:56
Example 3: Polynomial Function
17:34
Example 4: End Behavior
19:53
Analyzing Graphs of Polynomial Functions

33m 29s

Intro
0:00
Graphing Polynomial Functions
0:11
Example: Table and End Behavior
0:39
Location Principle
4:43
Zero Between Two Points
5:03
Example: Location Principle
5:21
Maximum and Minimum Points
8:40
Relative Maximum and Relative Minimum
9:16
Example: Number of Relative Max/Min
11:11
Example 1: Graph Polynomial Function
11:57
Example 2: Graph Polynomial Function
16:19
Example 3: Graph Polynomial Function
23:27
Example 4: Graph Polynomial Function
28:35
Solving Polynomial Functions

21m 10s

Intro
0:00
Factoring Polynomials
0:06
Greatest Common Factor (GCF)
0:25
Difference of Two Squares
1:14
Perfect Square Trinomials
2:07
General Trinomials
2:57
Grouping
4:32
Sum and Difference of Two Cubes
6:03
Examples: Two Cubes
6:14
Quadratic Form
8:22
Example: Quadratic Form
8:44
Example 1: Factor Polynomial
12:03
Example 2: Factor Polynomial
13:54
Example 3: Quadratic Form
15:33
Example 4: Solve Polynomial Function
17:24
Remainder and Factor Theorems

31m 21s

Intro
0:00
Remainder Theorem
0:07
Checking Work
0:22
Dividend and Divisor in Theorem
1:12
Example: f(a)
2:05
Synthetic Substitution
5:43
Example: Polynomial Function
6:15
Factor Theorem
9:54
Example: Numbers
10:16
Example: Confirm Factor
11:27
Factoring Polynomials
14:48
Example: 3rd Degree Polynomial
15:07
Example 1: Remainder Theorem
19:17
Example 2: Other Factors
21:57
Example 3: Remainder Theorem
25:52
Example 4: Other Factors
28:21
Roots and Zeros

31m 27s

Intro
0:00
Number of Roots
0:08
Not Nature of Roots
0:18
Example: Real and Complex Roots
0:25
Descartes' Rule of Signs
2:05
Positive Real Roots
2:21
Example: Positve
2:39
Negative Real Roots
5:44
Example: Negative
6:06
Finding the Roots
9:59
Example: Combination of Real and Complex
10:07
Conjugate Roots
13:18
Example: Conjugate Roots
13:50
Example 1: Solve Polynomial
16:03
Example 2: Solve Polynomial
18:36
Example 3: Possible Combinations
23:13
Example 4: Possible Combinations
27:11
Rational Zero Theorem

31m 16s

Intro
0:00
Equation
0:08
List of Possibilities
0:16
Equation with Constant and Leading Coefficient
1:04
Example: Rational Zero
2:46
Leading Coefficient Equal to One
7:19
Equation with Leading Coefficient of One
7:34
Example: Coefficient Equal to 1
8:45
Finding Rational Zeros
12:58
Division with Remainder Zero
13:32
Example 1: Possible Rational Zeros
14:20
Example 2: Possible Rational Zeros
16:02
Example 3: Possible Rational Zeros
19:58
Example 4: Find All Zeros
22:06
VII. Radical Expressions and Inequalities
Operations on Functions

34m 30s

Intro
0:00
Arithmetic Operations
0:07
Domain
0:16
Intersection
0:24
Denominator is Zero
0:49
Example: Operations
1:02
Composition of Functions
7:18
Notation
7:48
Right to Left
8:18
Example: Composition
8:48
Composition is Not Commutative
17:23
Example: Not Commutative
17:51
Example 1: Function Operations
20:55
Example 2: Function Operations
24:34
Example 3: Compositions
27:51
Example 4: Function Operations
31:09
Inverse Functions and Relations

22m 42s

Intro
0:00
Inverse of a Relation
0:14
Example: Ordered Pairs
0:56
Inverse of a Function
3:24
Domain and Range Switched
3:52
Example: Inverse
4:28
Procedure to Construct an Inverse Function
6:42
f(x) to y
6:42
Interchange x and y
6:59
Solve for y
7:06
Write Inverse f(x) for y
7:14
Example: Inverse Function
7:25
Example: Inverse Function 2
8:48
Inverses and Compositions
10:44
Example: Inverse Composition
11:46
Example 1: Inverse Relation
14:49
Example 2: Inverse of Function
15:40
Example 3: Inverse of Function
17:06
Example 4: Inverse Functions
18:55
Square Root Functions and Inequalities

30m 4s

Intro
0:00
Square Root Functions
0:07
Examples: Square Root Function
0:16
Example: Not Square Root Function
0:46
Radicand
1:12
Example: Restriction
1:31
Graphing Square Root Functions
3:42
Example: Graphing
3:49
Square Root Inequalities
8:47
Same Technique
9:00
Example: Square Root Inequality
9:20
Example 1: Graph Square Root Function
15:19
Example 2: Graph Square Root Function
18:03
Example 3: Graph Square Root Function
22:41
Example 4: Square Root Inequalities
25:37
nth Roots

20m 46s

Intro
0:00
Definition of the nth Root
0:07
Example: 5th Root
0:20
Example: 6th Root
0:51
Principal nth Root
1:39
Example: Principal Roots
2:06
Using Absolute Values
5:58
Example: Square Root
6:18
Example: 6th Root
8:40
Example: Negative
10:15
Example 1: Simplify Radicals
12:23
Example 2: Simplify Radicals
13:29
Example 3: Simplify Radicals
16:07
Example 4: Simplify Radicals
18:18
Operations with Radical Expressions

41m 11s

Intro
0:00
Properties of Radicals
0:16
Quotient Property
0:29
Example: Quotient
1:00
Example: Product Property
1:47
Simplifying Radical Expressions
3:24
Radicand No nth Powers
3:47
Radicand No Fractions
6:33
No Radicals in Denominator
7:16
Rationalizing Denominators
8:27
Example: Radicand nth Power
9:05
Conjugate Radical Expressions
11:47
Conjugates
12:07
Example: Conjugate Radical Expression
13:11
Adding and Subtracting Radicals
16:12
Same Index, Same Radicand
16:20
Example: Like Radicals
16:28
Multiplying Radicals
19:04
Distributive Property
19:10
Example: Multiplying Radicals
19:20
Example 1: Simplify Radical
24:11
Example 2: Simplify Radicals
28:43
Example 3: Simplify Radicals
32:00
Example 4: Simplify Radical
36:34
Rational Exponents

30m 45s

Intro
0:00
Definition 1
0:20
Example: Using Numbers
0:39
Example: Non-Negative
2:46
Example: Odd
3:34
Definition 2
4:32
Restriction
4:52
Example: Relate to Definition 1
5:04
Example: m Not 1
5:31
Simplifying Expressions
7:53
Multiplication
8:31
Division
9:29
Multiply Exponents
10:08
Raised Power
11:05
Zero Power
11:29
Negative Power
11:49
Simplified Form
13:52
Complex Fraction
14:16
Negative Exponents
14:40
Example: More Complicated
15:14
Example 1: Write as Radical
19:03
Example 2: Write with Rational Exponents
20:40
Example 3: Complex Fraction
22:09
Example 4: Complex Fraction
26:22
Solving Radical Equations and Inequalities

31m 27s

Intro
0:00
Radical Equations
0:11
Variables in Radicands
0:22
Example: Radical Equation
1:06
Example: Complex Equation
2:42
Extraneous Roots
7:21
Squaring Technique
7:35
Double Check
7:44
Example: Extraneous
8:21
Eliminating nth Roots
10:04
Isolate and Raise Power
10:14
Example: nth Root
10:27
Radical Inequalities
11:27
Restriction: Index is Even
11:53
Example: Radical Inequality
12:29
Example 1: Solve Radical Equation
15:41
Example 2: Solve Radical Equation
17:44
Example 3: Solve Radical Inequality
20:24
Example 4: Solve Radical Equation
24:34
VIII. Rational Equations and Inequalities
Multiplying and Dividing Rational Expressions

40m 54s

Intro
0:00
Simplifying Rational Expressions
0:22
Algebraic Fraction
0:29
Examples: Rational Expressions
0:49
Example: GCF
1:33
Example: Simplify Rational Expression
2:26
Factoring -1
4:04
Example: Simplify with -1
4:19
Multiplying and Dividing Rational Expressions
6:59
Multiplying and Dividing
7:28
Example: Multiplying Rational Expressions
8:36
Example: Dividing Rational Expressions
11:20
Factoring
14:01
Factoring Polynomials
14:19
Example: Factoring
14:35
Complex Fractions
18:22
Example: Numbers
18:37
Example: Algebraic Complex Fractions
19:25
Example 1: Simplify Rational Expression
25:56
Example 2: Simplify Rational Expression
29:34
Example 3: Simplify Rational Expression
31:39
Example 4: Simplify Rational Expression
37:50
Adding and Subtracting Rational Expressions

55m 4s

Intro
0:00
Least Common Multiple (LCM)
0:27
Examples: LCM of Numbers
0:43
Example: LCM of Polynomials
4:02
Adding and Subtracting
7:55
Least Common Denominator (LCD)
8:07
Example: Numbers
8:17
Example: Rational Expressions
11:03
Equivalent Fractions
15:22
Simplifying Complex Fractions
21:19
Example: Previous Lessons
21:36
Example: More Complex
22:53
Example 1: Find LCM
28:30
Example 2: Add Rational Expressions
31:44
Example 3: Subtract Rational Expressions
39:18
Example 4: Simplify Rational Expression
38:26
Graphing Rational Functions

57m 13s

Intro
0:00
Rational Functions
0:18
Restriction
0:34
Example: Rational Function
0:51
Breaks in Continuity
2:52
Example: Continuous Function
3:10
Discontinuities
3:30
Example: Excluded Values
4:37
Graphs and Discontinuities
5:02
Common Binomial Factor (Hole)
5:08
Example: Common Factor
5:31
Asymptote
10:06
Example: Vertical Asymptote
11:08
Horizontal Asymptotes
20:00
Example: Horizontal Asymptote
20:25
Example 1: Holes and Vertical Asymptotes
26:12
Example 2: Graph Rational Faction
28:35
Example 3: Graph Rational Faction
39:23
Example 4: Graph Rational Faction
47:28
Direct, Joint, and Inverse Variation

20m 21s

Intro
0:00
Direct Variation
0:07
Constant of Variation
0:25
Graph of Constant Variation
1:26
Slope is Constant k
1:35
Example: Straight Lines
1:41
Joint Variation
2:48
Three Variables
2:52
Inverse Variation
3:38
Rewritten Form
3:52
Examples in Biology
4:22
Graph of Inverse Variation
4:51
Asymptotes are Axes
5:12
Example: Inverse Variation
5:40
Proportions
10:11
Direct Variation
10:25
Inverse Variation
11:32
Example 1: Type of Variation
12:42
Example 2: Direct Variation
14:13
Example 3: Joint Variation
16:24
Example 4: Graph Rational Faction
18:50
Solving Rational Equations and Inequalities

55m 14s

Intro
0:00
Rational Equations
0:15
Example: Algebraic Fraction
0:26
Least Common Denominator
0:49
Example: Simple Rational Equation
1:22
Example: Solve Rational Equation
5:40
Extraneous Solutions
9:31
Doublecheck
10:00
No Solution
10:38
Example: Extraneous
10:44
Rational Inequalities
14:01
Excluded Values
14:31
Solve Related Equation
14:49
Find Intervals
14:58
Use Test Values
15:25
Example: Rational Inequality
15:51
Example: Rational Inequality 2
17:07
Example 1: Rational Equation
28:50
Example 2: Rational Equation
33:51
Example 3: Rational Equation
38:19
Example 4: Rational Inequality
46:49
IX. Exponential and Logarithmic Relations
Exponential Functions

35m 58s

Intro
0:00
What is an Exponential Function?
0:12
Restriction on b
0:31
Base
0:46
Example: Exponents as Bases
0:56
Variables as Exponents
1:12
Example: Exponential Function
1:50
Graphing Exponential Functions
2:33
Example: Using Table
2:49
Properties
11:52
Continuous and One to One
12:00
Domain is All Real Numbers
13:14
X-Axis Asymptote
13:55
Y-Intercept
14:02
Reflection Across Y-Axis
14:31
Growth and Decay
15:06
Exponential Growth
15:10
Real Life Examples
15:41
Example: Growth
15:52
Example: Decay
16:12
Real Life Examples
16:30
Equations
17:32
Bases are Same
18:05
Examples: Variables as Exponents
18:20
Inequalities
21:29
Property
21:51
Example: Inequality
22:37
Example 1: Graph Exponential Function
24:05
Example 2: Growth or Decay
27:50
Example 3: Exponential Equation
29:31
Example 4: Exponential Inequality
32:54
Logarithms and Logarithmic Functions

45m 54s

Intro
0:00
What are Logarithms?
0:08
Restrictions
0:15
Written Form
0:26
Logarithms are Exponents
0:52
Example: Logarithms
1:49
Logarithmic Functions
5:14
Same Restrictions
5:30
Inverses
5:53
Example: Logarithmic Function
6:24
Graph of the Logarithmic Function
9:20
Example: Using Table
9:35
Properties
15:09
Continuous and One to One
15:14
Domain
15:36
Range
15:56
Y-Axis is Asymptote
16:02
X Intercept
16:12
Inverse Property
16:57
Compositions of Functions
17:10
Equations
18:30
Example: Logarithmic Equation
19:13
Inequalities
20:36
Properties
20:47
Example: Logarithmic Inequality
21:40
Equations with Logarithms on Both Sides
24:43
Property
24:51
Example: Both Sides
25:23
Inequalities with Logarithms on Both Sides
26:52
Property
27:02
Example: Both Sides
28:05
Example 1: Solve Log Equation
31:52
Example 2: Solve Log Equation
33:53
Example 3: Solve Log Equation
36:15
Example 4: Solve Log Inequality
39:19
Properties of Logarithms

28m 43s

Intro
0:00
Product Property
0:08
Example: Product
0:46
Quotient Property
2:40
Example: Quotient
2:59
Power Property
3:51
Moved Exponent
4:07
Example: Power
4:37
Equations
5:15
Example: Use Properties
5:58
Example 1: Simplify Log
11:17
Example 2: Single Log
15:54
Example 3: Solve Log Equation
18:48
Example 4: Solve Log Equation
22:13
Common Logarithms

25m 23s

Intro
0:00
What are Common Logarithms?
0:10
Real World Applications
0:16
Base Not Written
0:27
Example: Base 10
0:39
Equations
1:47
Example: Same Base
1:56
Example: Different Base
2:37
Inequalities
6:07
Multiplying/Dividing Inequality
6:21
Example: Log Inequality
6:54
Change of Base
12:45
Base 10
13:24
Example: Change of Base
14:05
Example 1: Log Equation
15:21
Example 2: Common Logs
17:13
Example 3: Log Equation
18:22
Example 4: Log Inequality
21:52
Base e and Natural Logarithms

21m 14s

Intro
0:00
Number e
0:09
Natural Base
0:21
Growth/Decay
0:33
Example: Exponential Function
0:53
Natural Logarithms
1:11
ln x
1:19
Inverse and Identity Function
1:39
Example: Inverse Composition
1:55
Equations and Inequalities
4:39
Extraneous Solutions
5:30
Examples: Natural Log Equations
5:48
Example 1: Natural Log Equation
9:08
Example 2: Natural Log Equation
10:37
Example 3: Natural Log Inequality
16:54
Example 4: Natural Log Inequality
18:16
Exponential Growth and Decay

24m 30s

Intro
0:00
Decay
0:17
Decreases by Fixed Percentage
0:23
Rate of Decay
0:56
Example: Finance
1:34
Scientific Model of Decay
3:37
Exponential Decay
3:45
Radioactive Decay
4:13
Example: Half Life
5:33
Growth
9:06
Increases by Fixed Percentage
9:18
Example: Finance
10:09
Scientific Model of Growth
11:35
Population Growth
12:04
Example: Growth
12:20
Example 1: Computer Price
14:00
Example 2: Stock Price
15:46
Example 3: Medicine Disintegration
19:10
Example 4: Population Growth
22:33
X. Conic Sections
Midpoint and Distance Formulas

32m 42s

Intro
0:00
Midpoint Formula
0:15
Example: Midpoint
0:30
Distance Formula
2:30
Example: Distance
2:52
Example 1: Midpoint and Distance
4:58
Example 2: Midpoint and Distance
8:07
Example 3: Median Length
18:51
Example 4: Perimeter and Area
23:36
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
Radius
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
Example 2: Center and Radius
11:51
Example 3: Radius
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
Solving Quadratic Systems

47m 4s

Intro
0:00
Linear Quadratic Systems
0:22
Example: Linear Quadratic System
0:45
Solutions
2:49
Graphs of Possible Solutions
3:10
Quadratic Quadratic System
4:10
Example: Elimination
4:21
Solutions
11:39
Example: 0, 1, 2, 3, 4 Solutions
11:50
Systems of Quadratic Inequalities
12:48
Example: Quadratic Inequality
13:09
Example 1: Solve Quadratic System
21:42
Example 2: Solve Quadratic System
29:13
Example 3: Solve Quadratic System
35:02
Example 4: Solve Quadratic Inequality
40:29
XI. Sequences and Series
Arithmetic Sequences

21m 16s

Intro
0:00
Sequences
0:10
General Form of Sequence
0:16
Example: Finite/Infinite Sequences
0:33
Arithmetic Sequences
0:28
Common Difference
2:41
Example: Arithmetic Sequence
2:50
Formula for the nth Term
3:51
Example: nth Term
4:32
Equation for the nth Term
6:37
Example: Using Formula
6:56
Arithmetic Means
9:47
Example: Arithmetic Means
10:16
Example 1: nth Term
12:38
Example 2: Arithmetic Means
13:49
Example 3: Arithmetic Means
16:12
Example 4: nth Term
18:26
Arithmetic Series

21m 36s

Intro
0:00
What are Arithmetic Series?
0:11
Common Difference
0:28
Example: Arithmetic Sequence
0:43
Example: Arithmetic Series
1:09
Finite/Infinite Series
1:36
Sum of Arithmetic Series
2:27
Example: Sum
3:21
Sigma Notation
5:53
Index
6:14
Example: Sigma Notation
7:14
Example 1: First Term
9:00
Example 2: Three Terms
10:52
Example 3: Sum of Series
14:14
Example 4: Sum of Series
18:13
Geometric Sequences

23m 3s

Intro
0:00
Geometric Sequences
0:11
Common Difference
0:38
Common Ratio
1:08
Example: Geometric Sequence
2:38
nth Term of a Geometric Sequence
4:41
Example: nth Term
4:56
Geometric Means
6:51
Example: Geometric Mean
7:09
Example 1: 9th Term
12:04
Example 2: Geometric Means
15:18
Example 3: nth Term
18:32
Example 4: Three Terms
20:59
Geometric Series

22m 43s

Intro
0:00
What are Geometric Series?
0:11
List of Numbers
0:24
Example: Geometric Series
1:12
Sum of Geometric Series
2:16
Example: Sum of Geometric Series
2:41
Sigma Notation
4:21
Lower Index, Upper Index
4:38
Example: Sigma Notation
4:57
Another Sum Formula
6:08
Example: n Unknown
6:28
Specific Terms
7:41
Sum Formula
7:56
Example: Specific Term
8:11
Example 1: Sum of Geometric Series
10:02
Example 2: Sum of 8 Terms
14:15
Example 3: Sum of Geometric Series
18:23
Example 4: First Term
20:16
Infinite Geometric Series

18m 32s

Intro
0:00
What are Infinite Geometric Series
0:10
Example: Finite
0:29
Example: Infinite
0:51
Partial Sums
1:09
Formula
1:37
Sum of an Infinite Geometric Series
2:39
Convergent Series
2:58
Example: Sum of Convergent Series
3:28
Sigma Notation
7:31
Example: Sigma
8:17
Repeating Decimals
8:42
Example: Repeating Decimal
8:53
Example 1: Sum of Infinite Geometric Series
12:15
Example 2: Repeating Decimal
13:24
Example 3: Sum of Infinite Geometric Series
15:14
Example 4: Repeating Decimal
16:48
Recursion and Special Sequences

14m 34s

Intro
0:00
Fibonacci Sequence
0:05
Background of Fibonacci
0:23
Recursive Formula
0:37
Fibonacci Sequence
0:52
Example: Recursive Formula
2:18
Iteration
3:49
Example: Iteration
4:30
Example 1: Five Terms
7:08
Example 2: Three Terms
9:00
Example 3: Five Terms
10:38
Example 4: Three Iterates
12:41
Binomial Theorem

48m 30s

Intro
0:00
Pascal's Triangle
0:06
Expand Binomial
0:13
Pascal's Triangle
4:26
Properties
6:52
Example: Properties of Binomials
6:58
Factorials
9:11
Product
9:28
Example: Factorial
9:45
Binomial Theorem
11:08
Example: Binomial Theorem
13:48
Finding a Specific Term
18:36
Example: Specific Term
19:26
Example 1: Expand
24:39
Example 2: Fourth Term
30:26
Example 3: Five Terms
36:13
Example 4: Three Iterates
45:07
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Lecture Comments (7)

2 answers

Last reply by: DJ Sai
Mon Sep 17, 2018 8:46 PM

Post by Rita Semaan on January 7, 2013

For example one, when you were finding the first row and column you multiplied zero by zero then one by zero and added then said it was one. Shouldn't it be zero or did you multiply the one but the one on the second column?

3 answers

Last reply by: Dr Carleen Eaton
Tue Nov 29, 2011 8:56 PM

Post by Michael Fabrikant on January 22, 2011

In example three, why did the instructor multiply the bottom row by 1/3 and not negative 1/3?

Identity and Inverse Matrices

  • The identity matrix plays the role of the identity under multiplication.
  • The inverse of a matrix represents the inverse under multiplication.
  • A square matrix has an inverse if and only if its determinant is not zero.
  • The inverse of a 2 x 2 determinant can be calculated using a specific formula.

Identity and Inverse Matrices

Find the inverse of A if it exist [
− 10
4
5
− 1
]
  • Recall that to find the inverse A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ]
  • A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ] = [1/(( − 10( − 1) − (4)(5))][
    − 1
    − 4
    − 5
    − 10
    ] =
  • − [1/10][
    − 1
    − 4
    − 5
    − 10
    ] = [
    [1/10]
    [2/5]
    [1/2]
    1
    ]
[
[1/10]
[2/5]
[1/2]
1
]
Find the inverse of A if it exist [
3
0
3
0
]
  • Recall that to find the inverse A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ]
  • A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ] = [1/((3)(0) − (0)(3))][
    0
    0
    − 3
    3
    ] = [1/0] [
    − 1
    − 4
    − 5
    − 10
    ] = Inverse does not exist. You cannot divide by zero.
Inverse does not exist. You cannot divide by zero.
Find the inverse of A if it exist [
5
− 1
8
− 1
]
  • Recall that to find the inverse A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ]
  • A − 1 = [1/detA][
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ] = [1/((5)( − 1) − ( − 1)(8))][
    − 1
    1
    − 8
    5
    ] =
  • [1/3][
    − 1
    1
    − 8
    5
    ] = [
    − [1/3]
    [1/3]
    − [8/3]
    [1/3]
    ]
[
− [1/3]
[1/3]
− [8/3]
[1/3]
]
Find the inverse of A if it exist [
− 5
6
4
− 4
]
  • Recall that to find the inverse A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ]
  • A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ] = [1/(( − 5)( − 4) − (6)(4))][
    − 4
    − 6
    − 4
    − 5
    ] = − [1/4][
    − 4
    − 6
    − 4
    − 5
    ] = [
    1
    [3/2]
    1
    [5/4]
    ]
[
1
[3/2]
1
[5/4]
]
Find the inverse of A if it exist [
− 1
0
− 2
− 3
]
  • Recall that to find the inverse A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ]
  • A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ] = [1/(( − 1)( − 3) − (0)( − 2))][
    − 3
    0
    2
    − 1
    ] = [1/3][
    − 3
    0
    2
    − 1
    ] = [
    − 1
    0
    [2/3]
    − [1/3]
    ]
[
− 1
0
[2/3]
− [1/3]
]
Find the inverse of A if it exist [
− 1
− 8
1
8
]
  • Recall that to find the inverse A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ]
  • A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ] = [1/(( − 1)(8) − ( − 8)(1))][
    8
    8
    − 1
    − 1
    ] = [1/0][
    8
    8
    − 1
    − 1
    ] = Inverse does not exist.
Inverse does not exist.
Find the inverse of A if it exist [
− 1
− 2
3
10
]
  • Recall that to find the inverse A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ]
  • A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ] = [1/(( − 1)(10) − ( − 2)(3))][
    10
    2
    − 3
    − 1
    ] = −[1/4] [
    10
    2
    3
    − 1
    ]
−[1/4] [
10
2
3
− 1
]
Find the inverse of A if it exist [
5
0
− 6
0
]
  • Recall that to find the inverse A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ]
  • A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ] = [1/((5)(0) − (0)( − 6))][
    0
    0
    6
    5
    ] = [1/0] [
    0
    0
    6
    5
    ] = Inverse does not exist.
Inverse does not exist.
Find the inverse of A if it exist [
− 3
− 3
3
3
]
  • Recall that to find the inverse A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ]
  • A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b − c
    a
    ] = [1/(( − 3)(3) − ( − 3)(3))][
    3
    3 − 3
    − 3
    ] = [1/0][
    3
    3
    − 3
    − 3
    y ] = Inverse does not exist.
Inverse does not exist.
Find the inverse of A if it exist [
8
− 5
− 1
0
]
  • Recall that to find the inverse A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ]
  • A − 1 = [1/detA][
    d
    − b
    − c
    a
    ] = [1/(ad − bc)][
    d
    − b
    − c
    a
    ] = [1/((8)(0) − ( − 5)( − 1))][
    0
    5
    1
    8
    ] = −[1/5][
    0
    5
    1
    8
    ]
−[1/5][
0
5
1
8
]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Identity and Inverse Matrices

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Identity Matrix 0:13
    • Example: 2x2 Identity Matrix
    • Example: 4x4 Identity Matrix
    • Properties of Identity Matrices
    • Example: Multiplying Identity Matrix
  • Matrix Inverses 5:30
    • Writing Matrix Inverse
  • Inverse of a 2x2 Matrix 6:39
    • Example: 2x2 Matrix
  • Example 1: Inverse Matrix 10:18
  • Example 2: Find the Inverse Matrix 13:04
  • Example 3: Find the Inverse Matrix 17:53
  • Example 4: Find the Inverse Matrix 20:44

Transcription: Identity and Inverse Matrices

Welcome to Educator.com.0000

In today's lesson, we are going to continue on talking about matrices, this time focusing on two special types of matrices, identity and inverse matrices.0002

The identity matrix is a square nxn matrix, which has 1 for every element in the main diagonal, and 0 for every other element.0013

So, an identity matrix is a square matrix...and let's look at an example--for example, a 2x2 matrix that is an identity matrix.0026

It says that this has 1 for every element in the main diagonal; so that is 1 and 1; and 0 for every other element.0036

Looking at another example of this for a 4x4 matrix: in the main diagonal (that would be here), I have all 1's.0048

And I am going to fill in the other spots with 0's; so, down the main diagonal, I have 1's; everything else is a 0.0063

And this is 1, 2, 3, 4; 1, 2, 3, 4; so it is a 4x4 square matrix.0077

Now, let's talk about properties of these matrices.0083

For any nxn matrix A, if you multiply that matrix times its identity matrix, then you will get the original matrix back.0087

And you can also multiply these in either order.0097

So, in a way, when you think about identity--the identity property of multiplication--think back to regular numbers;0101

and when we talked about identity with multiplication, we would say that, if you multiplied the number n, any number,0112

times 1, you got that original number back; and that was the identity property with multiplication.0123

For example, if I took 3 times 1, I am going to get 3; or 10 times 1--I will get 10 back.0130

This is the same idea, but with matrices; so, this functions the same way as 1 in this case,0136

because if you multiply the matrix times its identity matrix, then you will get the original matrix back--0143

the same way as, if you multiplied a number times 1, you get the number back.0153

So, let's go ahead and try this--and remember that, unlike 1, though...it is just the number 1; but for identity matrices, there are multiple 1's.0157

You see that, for a 2x2 matrix, the identity matrix will be different than for a 4x4 or a 3x3.0164

OK, so let's use this 2x2, and let's say I had some matrix A, and it is going to be 3, -1, 2, 5.0170

And I am going to multiply it by its identity matrix, which is going to be this one, because it's a 2x2; and let's see what I get.0184

Well, recall that, for multiplication, if I am going to look for this position, row 1, column 1,0194

I am going to multiply row 1 of this first matrix times this first column here; so that is going to give me 3 times 1.0206

And then, I am going to find the sum of those products: 3 times 1, plus -1 times 0.0215

That is going to give me 3 plus 0, or 3; OK, so I am going to get 3 up here.0224

And let's look for row 1, column 2; it will be row 1, column 2; so 3 times 0, plus -1 times 1.0237

This is going to give me 0 - 1; 0 - 1 is -1, so -1 goes right here.0251

And continuing on now to the second row: row 2, column 1: row 2 of A, times column 1 of the identity matrix,0262

is going to give me 2 times 1, plus 5 times 0, which is going to equal...2 + 0 is 2.0274

OK, then finally, row 2, column 2: this is going to give me row 2, column 2--row 2 here, column 2 here.0287

Row 2 is 2 times 0, plus 5 times 1; well, that is simply 0 + 5, or 5.0297

So, you see that this property is shown here--that if I have a matrix A, and I multiply it by its identity matrix I, I get A back.0306

In the same way, for numbers, with the identity property of multiplication: if I take a number and multiply it by 1, I get the original number back.0317

OK, now talking about matrix inverses: if we have two nxn matrices (these are square matrices),0328

and we say that they are inverses of each other...if I have two nxn matrices, A and B, they are inverses of each other0339

if AB is equal to BA, and that product AB is equal to the identity matrix.0345

So, if the product of A and B turns out to be the identity matrix, then those two matrices are inverses of each other.0353

And the inverse of A, if it exists (and we are going to talk about that in a few minutes--that it may not always exist, and why),0368

is written as A-1: we pronounce this "A inverse."0374

OK, again, let's say I was given two matrices, A and B, and then I was asked, "Are they inverses?"0378

The way I would determine that is by multiplying those two out; and if the product is the identity matrix, then they are inverses.0389

If it is not, then those two are not inverses of each other.0395

Let's talk about finding the inverse of a 2x2 matrix.0399

If we have some matrix, A, that looks like this; the inverse of A, if it exists, is given by this formula: A-1 = 1/...0404

and if you look at this, ad - bc, this is going to look familiar; and that is the determinant of this matrix.0414

Remember that the determinant of a 2x2 matrix is given by the formula ad - bc; so it is 1 over the determinant, times this matrix.0423

And it is pretty easy to remember how you form this matrix, if you just look at it this way.0432

The way I got this is: I switched the positions of a and d, and then I took the opposite sign of b and the opposite sign of c.0438

So, if I had numbers here...let's look at that: if I was told that A is the matrix 3, 4, 1, 2, and I was told to find A-1:0449

well, then I would take 1 over the determinant, which is 3(2) - 4(1), times...0466

now, to find this matrix, I am going to switch these positions; I am going to put 2 here and 3 there.0481

I am going to take this number; and I keep the number, but I am going to put the opposite sign.0486

If this was a -4, I would have made it for; the same for the c position--I am going to keep it 1, but I am going to make it the opposite sign.0492

Now, stopping here for a second: I mentioned "if the inverse exists."0500

Looking at this, you can see that there could be a situation where the inverse does not exist.0505

OK, here I have 1/(6 -4); so that is 1/2; well, that is allowable.0511

But if this had been, say, 6 - 6, that would have been 0; that is not allowable--that is undefined.0518

So, I wouldn't have been able to find A-1.0524

OK, therefore, there are situations (and it is when this denominator turns out to be 0) that you can't find the inverse.0527

So, it is a good idea to look for this denominator right away, before you do any more work,0534

to determine if we even can find the inverse, so you don't waste any more time looking for something that does not exist.0539

Now that I have found that 1 over the determinant is actually 1/2,0546

I can just multiply using scalar multiplication (this is functioning as a scalar) to figure out A1.0552

So, I am just going to multiply 1/2 times each element: 1/2 times 2 is 2/2, which is 1.0562

And 1/2 times -4 is -4/2; that is -2; 1/2 times -1 is -1/2; and then, 1/2 times 3 is 3/2.0570

Now, if you wanted to check your work, you could always check your work by taking A times A-1,0584

and seeing if it comes out to the identity matrix, which it should if you did things correctly.0593

If I were to take 3, 4 times 1, 2, and multiply it by 1, -2, -1/2, 3/2, I would find that it does come out to the identity matrix for these 2x2 matrices.0598

OK, so first we are asked, in Example 1, to determine if A and B are inverses of each other.0617

So, are A and B inverses of each other? We are given these two matrices and asked if they are inverses.0625

Well, recall that if two matrices are inverses of each other, then if I multiply them,0631

I can actually multiply them in either order, and I am going to get the identity matrix.0637

So, before I proceed, let's just think about what the identity matrix is for a 2x2.0645

And I am going to have 1 along the main diagonal and 0 everywhere else, so this is the identity matrix in this situation.0649

I am going to go ahead and multiply these and see what I end up with.0654

OK, I am doing some matrix multiplication: first, row 1, column 1--first row, first column--that is 0 times 0, plus 1 times 1.0663

And that is going to give me 1 in this position.0684

Now, row 1, column 2: 0 times 0, plus...OK, then that would be 1, 1, times 2...row 1, column 2: I did 0 times 0, and now 1 times 2.0687

That is 0 plus 2, so that is 2; so in this position, I am going to get 2.0710

Now, row 2, column 1: 2 times 0, plus 1 times 1--that is going to give me 0 + 1 = 1 right here.0715

Row 2, column 2: 2 times 0, plus 1 times 2: 0 plus 2 is 2.0734

All right, so the question I was asked is if these are inverses of each other.0750

Well, when I took their product (the product of A and B), I found that AB does not equal the identity matrix.0755

This is not the identity matrix; this is; therefore, are A and B inverses of each other?0765

No, A and B are not inverses of each other.0770

And I was able to check that using this property.0781

Find the inverse if it exists: so finding the inverse of a 2x2 matrix, recall, uses this formula: 1 over the determinant,0786

which is ad - bc, times the matrix d, -b, -c, a.0796

A-1 (we are calling this matrix A) is 1 over the determinant.0809

Here I have 1 times 4, minus 2 times 3.0815

And just looking at this quickly, I see that this is 4 - 6, so that is -2; and therefore, that inverse is -1/2, so this inverse exists.0820

I didn't get 0 down here (before I proceed any farther).0832

That, times...da means I am going to switch these two: 4 will go in the a position; 1 will go in the d position.0836

For b, I am just going to take the opposite sign; and for c, I am just going to take the opposite sign.0845

Proceeding: this is going to give me 4 minus 6, as I said, which is going to give me -1/2, times 4, -2, -3, 1.0852

Now, I need to multiply each element in here by -1/2; and that is going to give me -1/2, times 4, which is -2;0862

-1/2 times -2 is -2/-2, which is 1; -1/2 times -3 is going to give me positive 3/2; and -1/2 times 1 is -1/2.0874

Now, if I wanted to check this, I could check it by saying that A times A-1 is the identity matrix.0893

This is A-1: let's go ahead and multiply this times A and see what happens.0900

So, let's try A times A-1, and just see what we get.0907

Here, I had A; that is 1, 2, 3, 4; and A-1, and that is -2, 1, 3/2, -1/2, equals...0913

OK, working over here, row 1, column 1: that is 1, and I am just going to go ahead and do part of this mentally...0937

1 times -2 is -2; 2 times 3/2...well, this cancels out; that just gives me 3.0951

This is going to give me -2 + 3 = 1; so I get 1 right here.0960

OK, row 1, column 2: that is 1 times 1 is 1; 2 times -1/2 is -2/2, which is -1.0965

Since 1 - 1 is 0, 0 goes in the row 1, column 2 position.0983

Let's go on to row 2, column 1: it is going to give me 3 times -2; that is -6.0990

And then, this is 4 times 3/2 equals 12/2; that equals 6.1009

So, -6 + 6 = 0; so I get 0 right here.1015

Row 2, column 2--the last position--equals: 3 times 1 is 3; 4 times -1/2 would be -4/2, or -2; 3 minus 2 is 1.1024

So you see, I actually did get the identity matrix; so I figured out A-1, and I was checking1046

that I was correct by saying, "Well, if this truly is the inverse, when I multiply these two together--1051

if those two matrices are inverses of each other, I will get the identity matrix."1057

And this is the identity matrix for a 2x2 matrix.1061

So, the inverse--the answer to the question they are asking--is this right here.1065

However, I checked my work right there.1070

Example 3: Find the inverse, if it exists.1074

I am using my formula, A-1 = 1 over the determinant, ad - bc, times...switching d and a and changing the signs of b and c.1079

So, A-1 equals 1 over...this is ad; that is -1 times 0, minus bc (minus -3 times -1).1095

Then, times this matrix...it was found by switching these two positions, 0 and -1.1110

Now, for -3, which is in the b position, I am going to take the opposite sign; that is going to be 3.1116

For -1, which is in the c position, I am going to take the opposite sign and make it 1.1122

OK, let's make sure that this is going to work now.1127

This is 0 minus -3 times -1; that is going to give me 3, so that is 1 over 0 minus 3; and there will be an inverse, because this is not 0.1131

If this had come out to be 0, I couldn't have found the inverse.1150

It is -1/3; 1 over 0 - 3 is -1/3...times this matrix.1154

This is going to give me, if I multiply -1/3 by each element in here...I am going to get -1/3 times 0, and that is going to give me 0.1164

Here, I have -1/3 times 3; that is going to give me -3/3, or -1, right here.1176

1/3 times 1 is 1/3; and 1/3 times -1 is -1/3.1187

So again, I was asked to find the inverse; I used this formula.1194

I first found the determinant, and I determined that it was -1/3; therefore, I could find the inverse.1198

And then, I multiplied it by the matrix, which consists of switching the positions of these two and reversing the signs of these two.1207

This becomes a positive; this also becomes a positive--we are giving them the opposite signs from what they had.1218

Then, I multiplied -1/3 by each element in this matrix to get this.1224

And if I were to check it (which I could), it would be by multiplying A (the original matrix) times its inverse.1229

And I would find that I get the identity matrix back.1238

OK, find the inverse if it exists: again, I am recalling my formula: A-1 is 1 over the determinant, ad - bc,1241

times the matrix found by switching a and d, and reversing the sign on b, and reversing the sign on c.1259

OK, so first the determinant: that is 1 times 6, minus 2 times 3.1269

The matrix: switch these two positions, and reverse the sign on 3, and reverse the sign there.1279

Now, you probably already saw that I didn't even need to go that far, because there is a problem.1286

What I have here is 1 times 6 (which is 6), minus 6, times its matrix.1291

Well, this turns out, obviously, to be 0; and since you can't have that--that is not allowed--it is undefined--1301

we can stop right here, because this is undefined.1311

In this case, the inverse does not exist; and the clue is: as soon as you see1317

that you are having to divide by 0, you know that you are not allowed to do that.1329

So, the inverse does not exist.1333

That concludes this lesson on Educator.com on identity and inverse matrices, and I will see you again soon!1338

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