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Raffi Hovasapian

Raffi Hovasapian

Determinants

Slide Duration:

Table of Contents

I. Linear Equations and Matrices
Linear Systems

39m 3s

Intro
0:00
Linear Systems
1:20
Introduction to Linear Systems
1:21
Examples
10:35
Example 1
10:36
Example 2
13:44
Example 3
16:12
Example 4
23:48
Example 5
28:23
Example 6
32:32
Number of Solutions
35:08
One Solution, No Solution, Infinitely Many Solutions
35:09
Method of Elimination
36:57
Method of Elimination
36:58
Matrices

30m 34s

Intro
0:00
Matrices
0:47
Definition and Example of Matrices
0:48
Square Matrix
7:55
Diagonal Matrix
9:31
Operations with Matrices
10:35
Matrix Addition
10:36
Scalar Multiplication
15:01
Transpose of a Matrix
17:51
Matrix Types
23:17
Regular: m x n Matrix of m Rows and n Column
23:18
Square: n x n Matrix With an Equal Number of Rows and Columns
23:44
Diagonal: A Square Matrix Where All Entries OFF the Main Diagonal are '0'
24:07
Matrix Operations
24:37
Matrix Operations
24:38
Example
25:55
Example
25:56
Dot Product & Matrix Multiplication

41m 42s

Intro
0:00
Dot Product
1:04
Example of Dot Product
1:05
Matrix Multiplication
7:05
Definition
7:06
Example 1
12:26
Example 2
17:38
Matrices and Linear Systems
21:24
Matrices and Linear Systems
21:25
Example 1
29:56
Example 2
32:30
Summary
33:56
Dot Product of Two Vectors and Matrix Multiplication
33:57
Summary, cont.
35:06
Matrix Representations of Linear Systems
35:07
Examples
35:34
Examples
35:35
Properties of Matrix Operation

43m 17s

Intro
0:00
Properties of Addition
1:11
Properties of Addition: A
1:12
Properties of Addition: B
2:30
Properties of Addition: C
2:57
Properties of Addition: D
4:20
Properties of Addition
5:22
Properties of Addition
5:23
Properties of Multiplication
6:47
Properties of Multiplication: A
7:46
Properties of Multiplication: B
8:13
Properties of Multiplication: C
9:18
Example: Properties of Multiplication
9:35
Definitions and Properties (Multiplication)
14:02
Identity Matrix: n x n matrix
14:03
Let A Be a Matrix of m x n
15:23
Definitions and Properties (Multiplication)
18:36
Definitions and Properties (Multiplication)
18:37
Properties of Scalar Multiplication
22:54
Properties of Scalar Multiplication: A
23:39
Properties of Scalar Multiplication: B
24:04
Properties of Scalar Multiplication: C
24:29
Properties of Scalar Multiplication: D
24:48
Properties of the Transpose
25:30
Properties of the Transpose
25:31
Properties of the Transpose
30:28
Example
30:29
Properties of Matrix Addition
33:25
Let A, B, C, and D Be m x n Matrices
33:26
There is a Unique m x n Matrix, 0, Such That…
33:48
Unique Matrix D
34:17
Properties of Matrix Multiplication
34:58
Let A, B, and C Be Matrices of the Appropriate Size
34:59
Let A Be Square Matrix (n x n)
35:44
Properties of Scalar Multiplication
36:35
Let r and s Be Real Numbers, and A and B Matrices
36:36
Properties of the Transpose
37:10
Let r Be a Scalar, and A and B Matrices
37:12
Example
37:58
Example
37:59
Solutions of Linear Systems, Part 1

38m 14s

Intro
0:00
Reduced Row Echelon Form
0:29
An m x n Matrix is in Reduced Row Echelon Form If:
0:30
Reduced Row Echelon Form
2:58
Example: Reduced Row Echelon Form
2:59
Theorem
8:30
Every m x n Matrix is Row-Equivalent to a UNIQUE Matrix in Reduced Row Echelon Form
8:31
Systematic and Careful Example
10:02
Step 1
10:54
Step 2
11:33
Step 3
12:50
Step 4
14:02
Step 5
15:31
Step 6
17:28
Example
30:39
Find the Reduced Row Echelon Form of a Given m x n Matrix
30:40
Solutions of Linear Systems, Part II

28m 54s

Intro
0:00
Solutions of Linear Systems
0:11
Solutions of Linear Systems
0:13
Example I
3:25
Solve the Linear System 1
3:26
Solve the Linear System 2
14:31
Example II
17:41
Solve the Linear System 3
17:42
Solve the Linear System 4
20:17
Homogeneous Systems
21:54
Homogeneous Systems Overview
21:55
Theorem and Example
24:01
Inverse of a Matrix

40m 10s

Intro
0:00
Finding the Inverse of a Matrix
0:41
Finding the Inverse of a Matrix
0:42
Properties of Non-Singular Matrices
6:38
Practical Procedure
9:15
Step1
9:16
Step 2
10:10
Step 3
10:46
Example: Finding Inverse
12:50
Linear Systems and Inverses
17:01
Linear Systems and Inverses
17:02
Theorem and Example
21:15
Theorem
26:32
Theorem
26:33
List of Non-Singular Equivalences
28:37
Example: Does the Following System Have a Non-trivial Solution?
30:13
Example: Inverse of a Matrix
36:16
II. Determinants
Determinants

21m 25s

Intro
0:00
Determinants
0:37
Introduction to Determinants
0:38
Example
6:12
Properties
9:00
Properties 1-5
9:01
Example
10:14
Properties, cont.
12:28
Properties 6 & 7
12:29
Example
14:14
Properties, cont.
18:34
Properties 8 & 9
18:35
Example
19:21
Cofactor Expansions

59m 31s

Intro
0:00
Cofactor Expansions and Their Application
0:42
Cofactor Expansions and Their Application
0:43
Example 1
3:52
Example 2
7:08
Evaluation of Determinants by Cofactor
9:38
Theorem
9:40
Example 1
11:41
Inverse of a Matrix by Cofactor
22:42
Inverse of a Matrix by Cofactor and Example
22:43
More Example
36:22
List of Non-Singular Equivalences
43:07
List of Non-Singular Equivalences
43:08
Example
44:38
Cramer's Rule
52:22
Introduction to Cramer's Rule and Example
52:23
III. Vectors in Rn
Vectors in the Plane

46m 54s

Intro
0:00
Vectors in the Plane
0:38
Vectors in the Plane
0:39
Example 1
8:25
Example 2
15:23
Vector Addition and Scalar Multiplication
19:33
Vector Addition
19:34
Scalar Multiplication
24:08
Example
26:25
The Angle Between Two Vectors
29:33
The Angle Between Two Vectors
29:34
Example
33:54
Properties of the Dot Product and Unit Vectors
38:17
Properties of the Dot Product and Unit Vectors
38:18
Defining Unit Vectors
40:01
2 Very Important Unit Vectors
41:56
n-Vector

52m 44s

Intro
0:00
n-Vectors
0:58
4-Vector
0:59
7-Vector
1:50
Vector Addition
2:43
Scalar Multiplication
3:37
Theorem: Part 1
4:24
Theorem: Part 2
11:38
Right and Left Handed Coordinate System
14:19
Projection of a Point Onto a Coordinate Line/Plane
17:20
Example
21:27
Cauchy-Schwarz Inequality
24:56
Triangle Inequality
36:29
Unit Vector
40:34
Vectors and Dot Products
44:23
Orthogonal Vectors
44:24
Cauchy-Schwarz Inequality
45:04
Triangle Inequality
45:21
Example 1
45:40
Example 2
48:16
Linear Transformation

48m 53s

Intro
0:00
Introduction to Linear Transformations
0:44
Introduction to Linear Transformations
0:45
Example 1
9:01
Example 2
11:33
Definition of Linear Mapping
14:13
Example 3
22:31
Example 4
26:07
Example 5
30:36
Examples
36:12
Projection Mapping
36:13
Images, Range, and Linear Mapping
38:33
Example of Linear Transformation
42:02
Linear Transformations, Part II

34m 8s

Intro
0:00
Linear Transformations
1:29
Linear Transformations
1:30
Theorem 1
7:15
Theorem 2
9:20
Example 1: Find L (-3, 4, 2)
11:17
Example 2: Is It Linear?
17:11
Theorem 3
25:57
Example 3: Finding the Standard Matrix
29:09
Lines and Planes

37m 54s

Intro
0:00
Lines and Plane
0:36
Example 1
0:37
Example 2
7:07
Lines in IR3
9:53
Parametric Equations
14:58
Example 3
17:26
Example 4
20:11
Planes in IR3
25:19
Example 5
31:12
Example 6
34:18
IV. Real Vector Spaces
Vector Spaces

42m 19s

Intro
0:00
Vector Spaces
3:43
Definition of Vector Spaces
3:44
Vector Spaces 1
5:19
Vector Spaces 2
9:34
Real Vector Space and Complex Vector Space
14:01
Example 1
15:59
Example 2
18:42
Examples
26:22
More Examples
26:23
Properties of Vector Spaces
32:53
Properties of Vector Spaces Overview
32:54
Property A
34:31
Property B
36:09
Property C
36:38
Property D
37:54
Property F
39:00
Subspaces

43m 37s

Intro
0:00
Subspaces
0:47
Defining Subspaces
0:48
Example 1
3:08
Example 2
3:49
Theorem
7:26
Example 3
9:11
Example 4
12:30
Example 5
16:05
Linear Combinations
23:27
Definition 1
23:28
Example 1
25:24
Definition 2
29:49
Example 2
31:34
Theorem
32:42
Example 3
34:00
Spanning Set for a Vector Space

33m 15s

Intro
0:00
A Spanning Set for a Vector Space
1:10
A Spanning Set for a Vector Space
1:11
Procedure to Check if a Set of Vectors Spans a Vector Space
3:38
Example 1
6:50
Example 2
14:28
Example 3
21:06
Example 4
22:15
Linear Independence

17m 20s

Intro
0:00
Linear Independence
0:32
Definition
0:39
Meaning
3:00
Procedure for Determining if a Given List of Vectors is Linear Independence or Linear Dependence
5:00
Example 1
7:21
Example 2
10:20
Basis & Dimension

31m 20s

Intro
0:00
Basis and Dimension
0:23
Definition
0:24
Example 1
3:30
Example 2: Part A
4:00
Example 2: Part B
6:53
Theorem 1
9:40
Theorem 2
11:32
Procedure for Finding a Subset of S that is a Basis for Span S
14:20
Example 3
16:38
Theorem 3
21:08
Example 4
25:27
Homogeneous Systems

24m 45s

Intro
0:00
Homogeneous Systems
0:51
Homogeneous Systems
0:52
Procedure for Finding a Basis for the Null Space of Ax = 0
2:56
Example 1
7:39
Example 2
18:03
Relationship Between Homogeneous and Non-Homogeneous Systems
19:47
Rank of a Matrix, Part I

35m 3s

Intro
0:00
Rank of a Matrix
1:47
Definition
1:48
Theorem 1
8:14
Example 1
9:38
Defining Row and Column Rank
16:53
If We Want a Basis for Span S Consisting of Vectors From S
22:00
If We want a Basis for Span S Consisting of Vectors Not Necessarily in S
24:07
Example 2: Part A
26:44
Example 2: Part B
32:10
Rank of a Matrix, Part II

29m 26s

Intro
0:00
Rank of a Matrix
0:17
Example 1: Part A
0:18
Example 1: Part B
5:58
Rank of a Matrix Review: Rows, Columns, and Row Rank
8:22
Procedure for Computing the Rank of a Matrix
14:36
Theorem 1: Rank + Nullity = n
16:19
Example 2
17:48
Rank & Singularity
20:09
Example 3
21:08
Theorem 2
23:25
List of Non-Singular Equivalences
24:24
List of Non-Singular Equivalences
24:25
Coordinates of a Vector

27m 3s

Intro
0:00
Coordinates of a Vector
1:07
Coordinates of a Vector
1:08
Example 1
8:35
Example 2
15:28
Example 3: Part A
19:15
Example 3: Part B
22:26
Change of Basis & Transition Matrices

33m 47s

Intro
0:00
Change of Basis & Transition Matrices
0:56
Change of Basis & Transition Matrices
0:57
Example 1
10:44
Example 2
20:44
Theorem
23:37
Example 3: Part A
26:21
Example 3: Part B
32:05
Orthonormal Bases in n-Space

32m 53s

Intro
0:00
Orthonormal Bases in n-Space
1:02
Orthonormal Bases in n-Space: Definition
1:03
Example 1
4:31
Theorem 1
6:55
Theorem 2
8:00
Theorem 3
9:04
Example 2
10:07
Theorem 2
13:54
Procedure for Constructing an O/N Basis
16:11
Example 3
21:42
Orthogonal Complements, Part I

21m 27s

Intro
0:00
Orthogonal Complements
0:19
Definition
0:20
Theorem 1
5:36
Example 1
6:58
Theorem 2
13:26
Theorem 3
15:06
Example 2
18:20
Orthogonal Complements, Part II

33m 49s

Intro
0:00
Relations Among the Four Fundamental Vector Spaces Associated with a Matrix A
2:16
Four Spaces Associated With A (If A is m x n)
2:17
Theorem
4:49
Example 1
7:17
Null Space and Column Space
10:48
Projections and Applications
16:50
Projections and Applications
16:51
Projection Illustration
21:00
Example 1
23:51
Projection Illustration Review
30:15
V. Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors

38m 11s

Intro
0:00
Eigenvalues and Eigenvectors
0:38
Eigenvalues and Eigenvectors
0:39
Definition 1
3:30
Example 1
7:20
Example 2
10:19
Definition 2
21:15
Example 3
23:41
Theorem 1
26:32
Theorem 2
27:56
Example 4
29:14
Review
34:32
Similar Matrices & Diagonalization

29m 55s

Intro
0:00
Similar Matrices and Diagonalization
0:25
Definition 1
0:26
Example 1
2:00
Properties
3:38
Definition 2
4:57
Theorem 1
6:12
Example 3
9:37
Theorem 2
12:40
Example 4
19:12
Example 5
20:55
Procedure for Diagonalizing Matrix A: Step 1
24:21
Procedure for Diagonalizing Matrix A: Step 2
25:04
Procedure for Diagonalizing Matrix A: Step 3
25:38
Procedure for Diagonalizing Matrix A: Step 4
27:02
Diagonalization of Symmetric Matrices

30m 14s

Intro
0:00
Diagonalization of Symmetric Matrices
1:15
Diagonalization of Symmetric Matrices
1:16
Theorem 1
2:24
Theorem 2
3:27
Example 1
4:47
Definition 1
6:44
Example 2
8:15
Theorem 3
10:28
Theorem 4
12:31
Example 3
18:00
VI. Linear Transformations
Linear Mappings Revisited

24m 5s

Intro
0:00
Linear Mappings
2:08
Definition
2:09
Linear Operator
7:36
Projection
8:48
Dilation
9:40
Contraction
10:07
Reflection
10:26
Rotation
11:06
Example 1
13:00
Theorem 1
18:16
Theorem 2
19:20
Kernel and Range of a Linear Map, Part I

26m 38s

Intro
0:00
Kernel and Range of a Linear Map
0:28
Definition 1
0:29
Example 1
4:36
Example 2
8:12
Definition 2
10:34
Example 3
13:34
Theorem 1
16:01
Theorem 2
18:26
Definition 3
21:11
Theorem 3
24:28
Kernel and Range of a Linear Map, Part II

25m 54s

Intro
0:00
Kernel and Range of a Linear Map
1:39
Theorem 1
1:40
Example 1: Part A
2:32
Example 1: Part B
8:12
Example 1: Part C
13:11
Example 1: Part D
14:55
Theorem 2
16:50
Theorem 3
23:00
Matrix of a Linear Map

33m 21s

Intro
0:00
Matrix of a Linear Map
0:11
Theorem 1
1:24
Procedure for Computing to Matrix: Step 1
7:10
Procedure for Computing to Matrix: Step 2
8:58
Procedure for Computing to Matrix: Step 3
9:50
Matrix of a Linear Map: Property
10:41
Example 1
14:07
Example 2
18:12
Example 3
24:31
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Lecture Comments (17)

1 answer

Last reply by: Professor Hovasapian
Fri May 15, 2015 1:32 AM

Post by Mark Freiheit on May 14, 2015

Isn't it true that every time you multiply a row by a -1 then add it to another row to reduce, you have to chance signs on the DET? It seems like you forgot this and just got lucky with the sign changes in the end.

1 answer

Last reply by: Professor Hovasapian
Sat Feb 1, 2014 4:31 PM

Post by Josh Winfield on January 31, 2014

Im having trouble mastering the concept of property #5;

5. If a single row or column is multiplied by a non zero constant, the det(A) is multiplied by r.

So then in the following example you have the matrix A = [1 2 3;1 5 3;2 8 6] and you go on to find the det(A) which i do agree with is definitely zero, but my mind tells me that the constants multiplying the det should be (1/2)(1/3) and not (2)(3). As we are technically multiplying the 3rd row but (1/2) and multiply the 3rd column by (1/3). Please advise on this.

Im very excited for this course in linear algebra, certainly after taking multivariate and vector cal with you last year and seeing your obvious ability to convey knowledge systematically and in an interesting way.

1 answer

Last reply by: Professor Hovasapian
Mon Nov 25, 2013 5:40 PM

Post by Eddy Noboa on November 22, 2013

Hello profesor, I have a question. At 19:50 when you are about to take the inverse of that determinant which is -2, at 19:52 you get out of the inverse 2,1,3/2,-1/2 which is (1/-2)* 1, then (1/-2)*2, then (1/2)*3 and finally (1/-2)*4. Doing it my self i do not get the same resuls as you. I get (1/-2)*1=1/-2, (1/-2)*2=-1, (1/-2)*2=-3/2, and (1/-2)*4=2/-1.Maybe im doing soemthing wrong if so can you please let me know. Thanks.

1 answer

Last reply by: Professor Hovasapian
Wed Oct 9, 2013 5:08 PM

Post by Mohamed Badawy on October 8, 2013

for the answer for the first example in minute 12, i used a different method and got the det (A) = -12.

I checked using several online math calcultors as well, I got the same answer of -12.

Now my question is, can there be more than one answer for the det of a matrix?

I understand how you got the determinant, but why didnt we get the same answer

0 answers

Post by Manfred Berger on June 27, 2013

Is there a way to express the determinant of a sum of matrices?

1 answer

Last reply by: Professor Hovasapian
Sat Dec 29, 2012 4:37 PM

Post by Eduardo Voloch on December 9, 2012

Can you create an example of a 4x4?

3 answers

Last reply by: Professor Hovasapian
Mon Oct 22, 2012 4:41 PM

Post by ran pashaely on October 21, 2012

I have been watch all the determinant part and I cant slove this problem :

Matrix A is regular (3x3)
B = 3A^-1
det=(A^T)=9
what is det(B)?

1 answer

Last reply by: Professor Hovasapian
Sun Jul 15, 2012 8:40 PM

Post by Saed Zahedi on December 29, 2011

Minute 2, detA = a11 a12 - a12 a21 should be
detA= a11 a22 - a12 a21

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
  • Determinants 0:37
    • Introduction to Determinants
    • Example
  • Properties 9:00
    • Properties 1-5
    • Example
  • Properties, cont. 12:28
    • Properties 6 & 7
    • Example
  • Properties, cont. 18:34
    • Properties 8 & 9
    • Example

Transcription: Determinants

Welcome back to educator.com, this is linear algebra, and today we are going to be talking about determinants.0000

Determinants have, determinants are very curious things in mathematics, obviously they play a very big role in linear algebra, but they also play a big role in other areas of mathematics.0007

Now we are not going to be necessarily be dealing with too many of the theoretical aspects, we are going to be more concerned with computation, using them to actually solve problems.0020

But it is good to know that determinants are very deep part of mathematical research, let's go ahead and get started.0031

Okay let us...0039

Take a matrix A, and we will take, we will use our notation A11, A12, A21, A22.0044

Okay, 2 by 2 matrix, we define the determinant of A...0057

... Another symbol for the determinant is a straight line up and do, we will be using that symbol, we will be using both interchangeable A12, A21, A22.0065

So it depends on what it is that you are talking about, you use the lines whether you want to specify the entries.0075

We use this more functional notation, determinant of this matrix A, when we want to simply speak about it in the abstract, so we define it as A11 times A12 - A12 times A21.0083

The pattern is this times that - this times that...0100

... Again a11 along the main diagonal this times that - that times this including the signs.0109

Let's do the same for a 3 by 3 and then we will do some examples, so let's say that B is our 3 by 3.0119

We have B11, B12, B13, B21, B22, B23, B31, B32, B33.0130

Linear algebra is notationally intensive, okay now, the determinant of B is equal to, okay I am going to write it out, and then we will talk about an actual pattern by that we can use.0149

B11 times B22, times B23 + B12 times B23, times B31 + B13 times B21 times B32...0173

... - B11 times B23 times B32 - B12, time B21, times B33...0200

... - B13 B22, B31,0222

Here is the pattern, there are different ways to think about this, and I know that many of you have of ‘course seen determinants before back in high school and perhaps in other areas of mathematics, perhaps in some college courses, in calculus or something like that.0232

Here is the general pattern, notice we have some that are +, +, +, + and some that are -, -, -...0247

... First one is B11, B22, B33, going from top left to bottom right, multiply everything down this way.0261

The second entry, just move over and go down again to the right, B12, times B23, but since you have nothing over here, just go down to this one, because you need three entries, notice each one of these has 3, 3, 3, 3, 3, 3.0271

You need three factors in the multiplication, so it's this times this times this.0289

Now you go to the next one B13, there's nothing here, but you need three of them, so you go here and here, so it's B13, B21 times B32, that takes care of the plus part.0295

Now let's deal with the minus part, go back to the B11, well B11 now go, try going down to the left, well there's nothing here at the left but you need three terms.0308

It's B11, B23, B32, go to the next one over, B12, B21, there's nothing here, but there is one here B33.0317

B13, B22, B31, there are different kind of pattern's that you can come up with, this is simply the best pattern that I personally have come up with to work with 3 by 3.0332

Again, you have probably seen determinants before, so whatever pattern you come up with is fine, I think this works out best, simply because you are going to the right, positive, you are going down the left, negative, if that makes sense.0347

Those have positive signs, when you are moving in this direction, you have negative signs, and of ‘course in this, you have a 3 by 3, each term has to have three things multiplied by each other, okay.0361

Lets do some examples...0372

Let's go back to, actually you know what?, I think I am going to try blue ink, let's define A as (1, 2, 3, 2, 1, 3, 3, 1, 2) okay, so let's do our pattern.0378

Let's see, let's go ahead and put something like that, and we will say the determinant of A, okay, 1 times 1 times 2 is 2, okay + 2 times 3 times 3.0397

2 times 3 is 6, 6 times 3 is 18 + 3 times 2 times 1, 6 okay.0412

Now, - 1 times 3 times 1 is 3, - 2 times 2 times 2, 2 times 2 is 4 + 4 is 8 + this 8 off - 3 times 1 times 3, -9.0423

When we add them all up, hopefully my arithmetic is correct, please check me you should get 6, so again positive this way, negative that way.0445

Let's do a 2 by 2, let's say B is equal to (4, -7, 2, -3) so now we have some negative entries, okay, the determinant of B is equal to this times this.0460

4 times -3 is -12 - this times this, this times this is -14, a - sign has to stay, so it's -12 - (-14) - 12 + 14, it is equal to 2.0479

Okay, so these signs here, this + here, + here, + here, -, -, -, they always stay there, it doesn't matter what these numbers are,0500

If this is negative, then a negative times a negative is positive, but you have to have three positive terms in a 3 by 3, you have to have 3 negative terms in a 3 by 3.0512

Negative in this case doesn't depends on what these numbers are, but those negative signs and these positive signs have to be there.0522

They are not part of the arithmetic; they are part of the definition of the determinant, okay.0529

Let's see here...0538

let's go over some properties of the determinants, just like we did properties of matrices, we will talk about some properties of determinants, so let A be an N by N matrix okay, then the determinant of the A transpose is the determinant of A.0543

In other words if I take A, take the transpose of it, and if I, then i take the determinant, it's the same as the determinant of A, no change, if you have a matrix and you interchange two rows or two columns this way or this way of the matrix, the determinant changes sign, so it goes from positive to negative, negative to positive.0559

Positive to negative, negative to positive, if two rows or columns of a matrix are equal, then the determinant equals 0, that isn't simple.0579

If a row or a column of A is entirely 0's then the determinant again is equal to 0, if a single row or a column is multiplied by a non-zero constant R, non-zero, then the determinant is multiplied by R.0593

The whole determinant is multiplied by R, if one row or column is just multiplied by r, let's do an example...0609

... We will let A = (1, 2, 3(1, 5, 3) and (2, 8, 6) okay, we want to find the determinant of A, in this case i am actually going to write it with this symbol, and you will see why in a minute.0621

I am going to rewrite it (1, 2, 3, 1, 5, 3, 2, 8, 6), in this case, using some of the properties particularly the one where we say if we multiply by a particular constant, I am going to use something that's going to be a Kent 2, factoring out, that you are used to from algebra, that's why I used this.0641

So I want you to see all of the entries, that's equal to... So notice this is (2, 8, 6).0663

You can divide this by 2, which means I can actually factor our a 2 from here, so i am going to put a 2, and i leave the other rows the same(1, 5, 3) and this becomes (1, 4, 3).0672

I can factor out a 3 here too, so I have 2, times 3,(1, 2, 1), I can factor out a 3 from this column, (1, 5, 1, 1, 4, 1) okay.0686

Now notice, I have a column of 1's and another column of 1's, two column that are the same, so now the determinant is equal to 2 times 3.0705

And when I have something, the two columns are the same, the determinant is 0, so saved myself a lot of problems, I didn't have to go through that whole strange, this diagonal, that diagonal, this entry, that entry, positive negative.0716

I used the properties of determinants, to actually make my life a lot easier, and I was able to find the determinants of the 3 by 3 pretty quickly, just by some standard algebraic manipulation, okay...0731

... Property number 6, if a multiple of one row or column of A, is added to another row or column, then the determinant is unchanged, remember that process of elimination that we did, we are doing Gauss Jordan reduction, Gauss Jordan elimination, where we multiplied some multiple of one row and added it to another.0750

When you do that, you are creating an equivalent system you remember, so the determinant doesn't change because the system is equivalent, now the seventh property, very important, if A is upper triangular, that means all entries below the main diagonal are 0, then the determinant a, of A is the product of the entries on the main diagonal.0771

A, upper triangular matrix, looks like let’s just say (1, 2, 3) let's say (3, 4, 6)...0792

... everything, so this is the main diagonal, okay, everything below the main diagonal is 0, so notice (0, 0, 0) but there are entries on the main diagonal, some of them can be 0's but they are not all 0's, okay.0807

Upper triangular means the upper part, the upper right hand art is the shape of a triangle, okay, when that's the case, then the determinant is just that.0820

It's kind of nice, so if you can actually turn it into buy a bunch of manipulation that you do to matrices, our elimination, interchanging rows, multiplying, you know one row by another, adding it to another row.0831

If you can actually change it to an upper triangular matrix and you just multiply those entries and you have your determinant, let's do an example, okay...0843

I have, I will go ahead and put my determinant symbol right there already (4, 3, 2) (3, -2, 5) and (2, 4, 6), okay i am going to go ahead and factor out a 2 from the third row right here.0858

That's equal to 2 times (4, 3, 2) (3, -2, 5), (1, 2, 3), okay.0879

I am actually going to switch this row, the third row and the first row, I am just going to switch them, and when I do that, I change the sign of the determinant, so I just take a -2 in front of that, and then I go (1, 2, 3), I will lead the second row the same (2, 5, 4, 3, 2) and these are pretty simple things that we are doing here.0892

Now, I have a 1 here and i have a 3 and then the 4, I am going to multiply this first row by -3 added to this, okay...0916

We end up with -2 times (1, 2, 3), (0, -8, -4) (4, 3, 2), now I am going to do the same thing with this row right here.0928

I am going to multiply this first row by -4, added to the third row, so I end up with -2 times (1, 2, 3), (0, -8, -4), (0, -5, -10), okay.0943

I am going to factor up, notice here, 4, there is an 8, there is a 4, I am going to go ahead and factor out a 4, so -2, we will take a 4 and I am going to also take out a 5 here, okay.0963

5, that turns it into (1, 2, 3)...0979

... (0, -2, -1) and (0, -1) oops, that is a -10, if I divide by 5, which should give me a -2.0986

Okay, so all I have done is factor up...1001

... I am actually going to do one more thing here, I am going to switch these rows simply because I want the 1 on top of the 2, personal choice, I don't necessarily need to do it, so again when I switch a row, I change the sign, so I get rid of that negative sign 2 times 4 times 5.1007

(1, 2, 3), (0, -1, -2), (0, -2, -1) okay, now I am going to multiply this by positive 2, this second row by +2 added to this one to get rid of this -2.1026

Okay, and when I do that, I get 2 times 4 times 5 (1, 2, 3), (0, -1, -2), (0, 0, 3), now have a...1045

... Upper triangular matrix, 0's entries along the main diagonal, now my determinant is equal to, 2...1063

... I am going to skip the parenthesis, times 4, times 5, times 1, times -1, times 3.1074

just this times, this times that, and now the determinant of this is just the entries along the main diagonal, and I have just a straight multiplication problem, and I should end up with -120, so properties, I have take a matrix...1086

... Subjected it to a bunch of, you know properties, simplified a little bit in order to find the determinant, okay...1102

... Few more properties to go, number 6 is if I take two matrices and I multiply them and then take the determinant, well it's the same as just taking the determinant of the first one times, the determinant of the second one, so determinant of A times B is equal to the determinant of A times determinant of B, reasonably straight forward there.1116

Now, if A is nonsingular, if it's invertible, if I have the inverse and if I take the determinant of it, it's the same as taking 1 over the determinant of the original matrix, notice this is not...1135

... Again we are not talking about 1 over A, 1 over A, A is a matrix, division by a matrix is not defined, but the determinant is a number, so division by a number is defined.1150

Let's go ahead and do an example, we will let A = (1, 2, 3, 4), we know that the determinant of A = 1 times 4, just 4 - 2 times 3, 6 = -2, okay.1162

Now, when we use our math's software to calculate the inverse of this matrix, we get the following (-2, 1, 3 halves, -1 half), when we take the determinant of the inverse, we get -2 times -1 half.1186

-2, I will write this one out, times -1 half, -...1210

... 1 times 3 halves, -2 times -1 half is a 1, -3 halves = -1 half.1219

Well, we said that the determinant of the inverse, of ‘course, we said that it's equal to, we want to confirm that, 1 divided by the determinant of the original.1233

Well the determinant of the inverse is -1 half, is it equal to 1 over -2, yes...1248

... If you want to know the determinant of the inverse, instead of finding the inverse and getting the determinant, you can just find the determinant of A and take the reciprocal of it.1260

Again, this is not 1 over the matrix A, this 1 over the determinant of A, the determinant is a number, the matrix itself is not a number.1267

And that covers determinants, thank you for joining us at educator.com, linear algebra; we will see you next time, bye, bye.1280

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