INSTRUCTORS  Carleen Eaton Grant Fraser  Dr. Carleen Eaton

Determinants

Slide Duration:

Section 1: 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
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
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
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
Section 2: 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
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
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
Section 3: 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
Section 4: 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
0:18
Same Dimensions
0:25
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
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
Section 5: Quadratic Functions and Inequalities

31m 48s

Intro
0:00
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

27m 3s

Intro
0:00
0:16
Standard Form
0:18
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

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
10:12
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
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

22m 48s

Intro
0:00
0:21
Standard Form
0:29
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
13:50
16:03
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

27m 5s

Intro
0:00
0:11
Test Point
0:18
0:29
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
Section 6: 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
0:13
Like Terms and Like Monomials
0:23
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
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
9:23
Odd Degrees
12:51
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
8:22
8:44
Example 1: Factor Polynomial
12:03
Example 2: Factor Polynomial
13:54
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
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
Section 7: 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
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
12:23
13:29
16:07
18:18

41m 11s

Intro
0:00
0:16
Quotient Property
0:29
Example: Quotient
1:00
Example: Product Property
1:47
3:24
3:47
6:33
7:16
Rationalizing Denominators
8:27
9:05
11:47
Conjugates
12:07
13:11
16:12
16:20
16:28
19:04
Distributive Property
19:10
19:20
24:11
28:43
32:00
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
19:03
Example 2: Write with Rational Exponents
20:40
Example 3: Complex Fraction
22:09
Example 4: Complex Fraction
26:22

31m 27s

Intro
0:00
0:11
0:22
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
11:27
Restriction: Index is Even
11:53
12:29
15:41
17:44
20:24
24:34
Section 8: 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

55m 4s

Intro
0:00
Least Common Multiple (LCM)
0:27
Examples: LCM of Numbers
0:43
Example: LCM of Polynomials
4:02
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
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
Section 9: 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
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
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
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
16:54
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
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
Section 10: 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
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
11:51
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

47m 4s

Intro
0:00
0:22
0:45
Solutions
2:49
Graphs of Possible Solutions
3:10
4:10
Example: Elimination
4:21
Solutions
11:39
Example: 0, 1, 2, 3, 4 Solutions
11:50
12:48
13:09
21:42
29:13
35:02
40:29
Section 11: 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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• ## Related Books 2 answersLast reply by: Andrew LiuThu Jun 4, 2020 6:47 PMPost by DJ Sai on September 10, 2018Why do you still answer these questions? 1 answer Last reply by: Dr Carleen EatonTue Nov 18, 2014 4:22 PMPost by Adnaan Akbarali on November 3, 2014Dr Carleen You are amazing ! thank you for your time to help us ! 3 answers Last reply by: Dr Carleen EatonMon Nov 7, 2011 8:45 PMPost by Charles Hassler on October 21, 2010isn't bc (0*-1)= 0 so wouldn't the determinant in the 2x2 example be 6?

### Determinants

• A determinant is a value associated with a square matrix.
• Third-order determinants can be evaluated by expanding by minors along any row or column of the associated matrix.
• These determinants can also be found by calculating products of entries along the diagonals and then adding and subtracting these products.

### Determinants

Find the determinant of [
 − 2
 5
 − 3
 5
]
• To find the determinant, |
 a
 b
 c
 d
• det[
 − 2
 5
 − 3
 5
] = |
 − 2
 5
 − 3
 5
| =
2Find the determinant of
• To find the determinant,
3Find the determinant of
• To find the determinant,
4Find the determinant of
• To find the determinant,
• enditemize
5Find the determinant of
• To find the determinant,
6Find the determinant of using the Expansion by minors Method.
• Recall that Expansion by minors means the following
7Find the determinant of using the Expansion by minors Method
• Recall that Expansion by minors means the following
8Find the determinant of using the Expansion by minors Method
• Recall that Expansion by minors means the following
9Find the determinant of using the Diagonal Method
• When using the diagonal method, copy the first two columns next to the third column.
• Start at the upper left and find the sum of the products of the diagonals.
• Next, start at the lower left and subtract products of the diagonals
• Simplify
10Find the determinant of using the Diagonal Method
• When using the diagonal method, copy the first two columns next to the third column.
• Start at the upper left and find the sum of the products of the diagonals.
• Next, start at the lower left and subtract products of the diagonals
• Simplify

*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.

### Determinants

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
• What is a Determinant 0:13
• Square Matrices
• Vertical Bars
• Determinant of a 2x2 Matrix 1:21
• Second Order Determinant
• Formula
• Example: 2x2 Determinant
• Determinant of a 3x3 Matrix 2:50
• Expansion by Minors
• Third Order Determinant
• Expanding Row One
• Example: 3x3 Determinant
• Diagonal Method for 3x3 Matrices 13:24
• Example: Diagonal Method
• 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

### Transcription: Determinants

Welcome to Educator.com.0000

In today's lesson, we are going to cover determinants.0002

And determinants are associated with matrices, and they are useful, because they can help you solve problems such as systems of equations.0004

OK, first of all, defining a determinant--what is a determinant?0014

Every square matrix has a number associated with it, called the determinant of the matrix.0017

And notice that this is limited to square matrices--matrices with the same number of rows and columns.0022

And just talking about some convention: when we write a determinant, we use vertical lines, like absolute value bars.0030

For example, if I had a square matrix that was 2x2, and I wanted to indicate0041

that I am talking about the determinant of that matrix, I would write it as such.0046

So again, this is for square matrices only; and you use these vertical lines to denote that you are working with the determinant,0054

rather than the matrix (this is the determinant), because if I just used these brackets, then that would actually be the 2x2 square matrix.0062

All right, starting out, we are finding determinants of 2x2 matrices.0081

Given the matrix with the elements a, b, c, d, the determinant written with these vertical bars, a, b, c, d, can be found using this formula.0087

And a determinant that is associated with a 2x2 matrix is called a second-order determinant.0097

So, this formula, ad - bc (you multiply these two elements and then subtract the product of these two elements) is pretty straightforward.0105

Let's try an example, using the 2x2 matrix.0118

To find this determinant, I am going to multiply 2 times 3; and from that product, I will subtract this product, BC.0127

So, I have ab, and I am going to then find bc, which is 0, -1; this equals 6 - -1, which is a negative and a negative; that gives me a positive 7.0140

Therefore, this second-order determinant (for a 2x2 matrix) is 7.0156

Finding the determinant of a 3x3 matrix is more complex, and there are a couple of methods.0171

In certain cases, one method works better; in others, the other works better.0177

Or you may just have a preference for one method over the other; and I am going to show you both of these methods.0181

And this first one is called expansion by minors: so, I am looking here, and it is asking me for a third-order determinant.0187

So, it is for a 3x3 matrix, so the determinant here is a third-order determinant.0198

And one way to find it is to do what is called expansion by minors, and break this down into three second-order determinants.0203

Once you get these second-order determinants...we already talked about finding a second-order determinant.0212

If you had one that was a, b, c, d, you would find that by taking ad, and then subtracting the product bc from it.0226

So, if I can get these into second-order determinants, then it is pretty easy to work with.0239

Now, looking at it, actually what is happening here is: what we did is something called expanding along the first row (expanding along row 1).0246

You can expand along any row; but right here, what they did is expanded along row 1.0258

And the reason I know that is: I look here, and I see a (that is from row 1), b, c.0263

The way you find these minors is: you find what you are working with (which is a), and you eliminate the column that a is in, and the row that a is in.0270

And that leaves you with e, f, h, i.0286

OK, looking at what happened here with b: for b, I eliminated the column that b is in, which would be this one,0291

and the row that b is in, which would leave me d, g, f, i, right here.0304

For c, eliminate the column that c is in and the row that c is in; and that is going to leave me with this: d, e, g, h.0312

Now, you notice that there are alternating positive and negative signs.0325

And it is important to understand how this works, because you can expand along other rows, and then you need to know what to make the sign.0329

So, it starts out positive; a is positive; b is negative; here, c is going to be positive; and then, you just alternate.0337

Since I was working with a right here, the associated sign is positive; for b, it is negative; and so on.0348

You can actually expand along any row; and you will see, in a little while, why you may want to choose another row.0355

You want to pick the row that will give you the situation that is easiest to work with, in terms of finding these determinants and multiplying.0363

If I were to expand along row 2, then I would take whatever is here first,0371

and I would eliminate this column and this row; and then I would have these four things left.0377

Then, I would take this second element; I would cross out this column and this row and take what is left; and so on.0386

Let's do an example of expansion by minors.0395

All right, the first example: 0, 2, 1, 2, 3, 2, 1, 2, 0; I need to find this third-order determinant using expansion by minors.0401

And I am going to expand along the first row, just to keep it simple, like we did up here.0416

So, I start out with this 0, and I know that there is a positive sign associated with that.0422

And I am going to end up with looking for this second-order determinant.0428

So, 0 is in this first column (so I cross it out) and this first row; that is going to leave me with 3, 2, 2, 0.0434

Now, next I am rewriting this and erasing this, so I can cross out again: 0, 2, 1...0450

The second section right here is going to be associated with this 2, and it is going to be negative: -2 times this second-order determinant.0467

2 is in the first row, second column; cross that out; that is going to leave me with 2, 2, 1, 0.0481

Next, I have a 1 here, and there is a positive sign associated with that; so it is + 1, and that is a second-order determinant.0496

And I am going to find that (let's reconstruct this: 0, 2, 3, 2)...so, my 1 is right here in this first row, third column.0508

Cross out the first row and the third column to leave me 2, 3, 1, 2.0527

Now, this is much easier to work with: and one thing that is really nice is that there is a 0 here.0535

What I need to do is find each of these determinants, and then multiply them by the scalar, the number that is right out here.0542

OK, well, no matter what this turns out to be, I am multiplying it by 0; so it is just going to drop out.0549

So, I am going to have 0 times something; and it doesn't really matter what it is that is going to drop out.0558

Then, I am going to have -2 times this determinant; so I am going to find this determinant, using ad - bc.0563

ad is 2 times 0, minus bc--that is minus 2 times 1; plus...here is 1, times ad (which is 2 times 2), minus bc.0571

Simplifying this: this is just 0 times whatever was in here--it didn't matter, because 0 times anything is 0;0620

minus 2 times...2 times 0 is 0, minus 2 times 1, plus 1 times...2 times 2 is 4, minus 3 times 1, which is 3.0628

This equals 0 minus 2 times -2, plus 1 times 4 - 3, which is 1.0645

Now, just coming up here to finish this out: get rid of this 0 at this point: this is -2 times -2, which is 4; 1 times 1 is 1; so this is 5.0665

So, at last, we found this third-order determinant for this, and it is 5.0686

Just taking it step-by-step, I expanded along row 1; I found the minor associated with each of these numbers0694

by crossing out the column and the row that that number was in, and writing out the four numbers (elements) that were left.0702

Then, I multiplied by 0 for this first one; for the second one, I crossed out row 1, column 2.0711

I was left with 2, 2, 1, 0, and multiplied that by 2, but I made it a -2, because of the negative signs associated with this.0720

For the third one, I had 1; I crossed out row 1, column 3; I was left with 2, 3, 1, 2.0729

I multiplied by a positive 1 (because it has a positive sign associated with that).0736

Now, I mentioned that you might want to expand along a different row.0739

The best rows to expand along are ones with 0's; and if there had been a row with two 0's, I would have expanded along that row.0743

And you see why: I didn't have to do any work here, because this is going to drop out.0751

If the second row, for example, would have had two 0's, I would have chosen that row.0754

I would have done the same method and expanded along row 2.0758

And if I had had two 0's there, great--this drops out; that drops out; I would only have had to work with this one.0762

So, you choose the row to expand out that has the most 0's, because it will be the easiest one to work with.0768

This is one method of finding a third-order determinant; and once I get these,0775

I find the second-order determinants using what we talked about before, this formula ad - bc.0779

I knew the first one was just going to drop out; the second one is represented right here; it is -2 times ad - bc.0785

And the third one is 1 times ad - bc.0795

Then, I just did the math to get 5.0801

Now, there is another method we are going to learn today for working with 3x3 matrices when you are finding the determinant.0803

And it is called the diagonal method: using an example to illustrate this, if you need to find0812

this third-order determinant, and you decide you want to use the diagonal method;0829

the first step is to rewrite the first two columns of numbers, right next to the determinant that you are looking for.0833

So, I am starting out with this; and I am going to just copy these two columns over.0854

It is 3, -1, 0, 1, 2, -2: that is the first step.0858

Next, you begin at the upper left, and you go along the diagonals, going from upper-left to down.0866

And you find the sum of the product of the diagonals.0877

Start at upper left, and find the sum of the products of the diagonals.0886

Now, what does that mean? It is actually simpler than it sounds.0903

Here, I have this first diagonal, 3, 2, 3; I am going to find that product, 3 times 2 times 3.0907

And then, I want to add it--I want to find the sum with the next diagonal, starting at the upper left: 1, 4, 0...1 times 4 times 0.0914

Then, I am going to add that to the next diagonal, 0, -1, times -2.0928

Then, you start at the lower left, and you find the products of those diagonals and subtract those.0939

Start at lower left, and subtract products of the diagonals.0948

Starting at the lower left, I have 0, 2, 0 on this diagonal--so minus 0 times 2 times 0.0973

On the next diagonal, I am going to subtract (just to make this clearer, let's go like this): minus this next diagonal, which is -2, 4, and 3.0986

Then, I am going to subtract this next diagonal: 3 times -1 times 1.1005

OK, so all you are going to do is find the products of the diagonals, starting up here and going down; find those products: 1, 2, 3; add them.1013

Find the products of these diagonals going up; subtract them.1024

All right, just doing the math: 3 times 2 times 3 is 9 times 2; that is 18; plus 1 times 4 times 0...0, plus 0 times -1 times -2...0,1030

minus 0 times 2 times 0 is 0; minus -2 times 4 (is 8) times 3 (is -24); minus 3 times -1 times 1 (so that is just -3).1046

I am simplifying things a bit: that is 18; let's get rid of these 0's--we don't need those; and this is minus -24, so I am going to write that as + 24.1067

Minus -3 is plus 3; 18 + 24 + 3 is 45.1077

If you would have used the other method, expansion by minors, you would have gotten the same result of 45 as this third-order determinant.1086

But you need to decide which method you like better and, given the situation, which method seems easier.1094

So again, in order to use diagonals to find this third-order determinant, we have to first rewrite the first two columns of numbers right next to the determinant.1100

That allowed us to form sets of diagonals; and we took the downward-going diagonals from here,1110

starting up at the upper left, and going down; and we found the products of each of those and added those.1117

Then, we started at the lower left, and found the products of those diagonals, and subtracted them.1123

Working out that math gave me 45, so that is the third-order determinant.1130

In this first example, we are going to find the second-order determinant.1142

If we have a second-order determinant with elements a, b, c, d, the determinant can be found1151

by multiplying ad--finding that product--and subtracting bc.1157

All right, so doing that right here is going to give me 1 times 4, minus b times c, which is -1.1162

OK, this equals 4 minus -3; if I have a negative and a negative, that gives me a positive: 4 plus 3, so it is 7.1181

So, the determinant here is 7, using this formula.1193

So, the second-order determinant...you just need to follow this formula--it is pretty straightforward.1197

A third-order determinant: I have two choices--I can either use diagonals, or I can use expansion of the minors.1204

And I am actually going to expand; and I am going to choose which row I should expand along.1211

And recall that I said, if you have 0's, to go to that row, because then things cancel out, and it makes your life a lot easier.1218

So, looking right here, I have a good situation, because I actually have two 0's up here in this row.1225

So, I am going to choose to expand along row 1.1232

Now, recall that, before we even do anything else, the signs associated would be positive, negative, positive;1236

negative, positive, negative...just so we make sure we are using the right signs for the row we are expanding along.1247

All right, so first, I am going to work with the 0.1252

And it doesn't matter if it is positive or negative, because it is 0; but it would have been positive.1255

Now, to find that associated minor, I am going to cross out the row that it is in and the column that it is in, so row 1 and column 1 are gone.1260

That leaves me with 3, 4, 2, 5; OK, I am rewriting this, just to make it really clear what is happening.1270

Now, next I am working with this 0; it is a 0, so it doesn't matter; but had it been another number,1285

it would have mattered that this would have been a negative sign associated with it.1291

So, right here, I am going to write the negative 0; and then, I am going to find the minor associated with it.1295

OK, this 0 is in row 1, column 2; that leaves me with 2, 4, -1, 5.1308

Next, I am writing this again, just so you can see each step.1322

The next number I am working with is 1; and that is going to be associated with a plus sign, because there is a positive sign right there in that position.1332

It's times this minor, which is going to be...I am going to cross out row 1, because that is where the 1 is, and column 3, to leave me with 2, 3, -1, 2.1340

OK, so as you can see, these 0's make the math much easier, because 0 times whatever this turns out to be is...1355

if I had bothered to find this determinant, it doesn't matter what it is, because it is just going to drop out.1367

The same with this one--I don't know what it is, and it doesn't actually matter.1373

Now, I need to find this determinant; and remember that the way I do it is: given that these are the positions,1377

a, b, c, d, I am going to find the product of a and d; and from that, I will subtract the product of b and c.1385

So, this is plus 1 times this; 2 times 2, minus 3 times -1.1394

This is 0; it drops out; 0 drops out; all I am working with is this.1417

1 times 2 times 2 is 4, minus 3 times -1 (that is -3), equals 1 times 4; a negative and a negative gives me a positive, so that is 1 times 7, which equals 7.1421

OK, so in Example 2, here we found the third-order determinant to be 7.1438

And I did that by expanding the minors along row 1; and I chose row 1 because I saw 0's.1444

So, I expanded by eliminating the column and row that that first 0 was in, leaving me 3, 4, 2, 5; and that was associated with a plus.1450

For this next number, it was a 0, and I saw that there was a negative associated with that, and a 0.1466

And then, I crossed out the row and the column that 0 was in, and was left with 2, 4, -1, 5; I wrote those there.1473

For the third element here, which was 1, I saw a positive associated with that, so I put + 1.1484

I crossed out the first row and the third column, and was left with 2, 3, -1, 2.1493

Then, once I had all this, I had to find these determinants.1499

And then, I realized that, since I am multiplying this by 0, I don't need to waste time even finding that; it is just going to drop out.1504

The same with this one; for this one, I used the formula to find this second-order determinant, which is ad - bc.1510

So, I have 0, 0, plus 1, times ad, which is 2 times 2, minus 3 times -1.1519

I calculated that out to find out that it is 7; the result is 7.1530

For this third-order determinant, I am going to use the diagonals method.1537

Recall that, with the diagonal method, the first thing you need to do is copy over those first two columns of numbers right next to the determinant.1542

You copy those over to expand the array; and then, you are going to find the products of diagonals,1555

beginning at the upper left: this diagonal, this diagonal, and this diagonal.1565

And you are going to add those products.1570

OK, so first, I have 2 times 4 times 1; then, I am going to add that to the product of this next diagonal, which is 3 times 6 times 0.1572

For the next diagonal, I have 1 times -1 times 8; OK, so that took care of those diagonals.1589

Then, I am going to start at the lower left, down here; and I am going to find the product of those diagonals; and this time, I am going to subtract.1599

So, minus 0 times 4 times 1; then, I am going to subtract the product of this next diagonal, and the product is the product of 8, 6, and 2.1606

Then, the next diagonal: subtract that product, which is the product of 1 times -1 times 3.1630

OK, then it is just working out all the arithmetic and making sure that you don't make any errors.1640

2 times 4 times 1 gives me 8, plus 3 times 6 (that is 18); then, this is...let's see...1 times -1 is -1, times 8--that gives me -8.1645

Actually, it is 3 times 6 times 0; correction--that is 0: 3 times 6 times 0--that is 0.1671

This is 1 times -1, is -1; times 8--that is -8.1678

Here, again, I have a 0; so this is just going to become 0.1683

Here, I have minus 8 times 6, which is 48, times 2--that is going to give me 96, minus 1 times -1 is -1, times 3 is -3.1690

Simplifying this a little further, this is 8; eliminate the 0; minus 8; eliminate the 0; minus 96; and this is minus -3, so it is plus 3.1710

8 minus 8 is 0, so that leaves me with -96 plus 3, to give me -93; so, this determinant is -93.1722

And I used the diagonal method, first copying these first two columns of numbers right here,1733

then finding the products of the diagonals, going from upper left to lower right, and adding those products.1740

Then, I went to the lower left, and along those diagonals, I found the products; and I subtracted those, did my arithmetic, and got -93 as the determinant.1749

OK, in Example 4, again, I am looking for a third-order determinant.1763

And you could use either method; I am going to actually expand along row 1.1770

That has a 0 in it, so I am going to use expansion of the minors.1779

You could have also used row 2 or 3; they all have 0's in them; but these also have smaller numbers, so I am going to go with row 1.1783

Now, with the expansion of the minors, I want to keep track of my positive and negative signs.1790

I am going to start with 0, and that is associated with a positive.1800

And when I expand that, I am going to eliminate this first column and this first row; and it is going to leave me with 3, 0, 6, 3.1805

Then, next, I am going to rewrite this...2, 3, 4...so that you can see very clearly.1819

The next number I am working with is 2, and that is associated with a negative sign; so I am going to write -2.1827

It is in row 1, column 2; get rid of those, and see that I am left with 4, 0, 0, 3.1832

Next, I am rewriting this original; I am now working with the 3, and that has a positive sign associated with it, so plus 3.1844

Eliminate the row it is in and the column it is in; 4, 3, 0, 6.1859

Now, I have 0 times this determinant; that is just going to drop out, so I am not even going to waste my time figuring out that determinant.1868

These other ones...I use ad - bc to find them.1878

So, I have minus 2, times...this is 4 times 3, minus 0 times 0, plus (so I have this taken care of) 3 times ad - bc.1887

So, that is 4 times 6, minus 0 times 3; OK, do the math now.1906

This is just 0; it is going to drop out; minus 2, times 12 minus 0, plus 3 times 24 minus 0.1914

This becomes 0 minus 2, times 12, plus 3 times...this is just 24.1925

Let's get rid of the 0 at this point: -2 times 12 is -24; 3 times 24 is 72; -24 + 72 = 48.1935

OK, so we found this third-order determinant by using expansion of the minors along row 1,1949

and being careful to keep the signs straight on the numbers associated with those,1955

and crossing out the row and column for each number to find the associated minor.1961

Then, once I had those, I saw that this first one was just going to drop out.1969

For the second and third, I used ad - bc to figure this out, and multiplied this one by -2 and this one by 3; working that out gave me 48.1973

That concludes this session of Educator.com on determinants; I will see you next lesson!1988

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