For more information, please see full course syllabus of Linear Algebra

For more information, please see full course syllabus of Linear Algebra

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### Cofactor Expansions

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
- Cofactor Expansions and Their Application 0:42
- Cofactor Expansions and Their Application
- Example 1
- Example 2
- Evaluation of Determinants by Cofactor 9:38
- Theorem
- Example 1
- Inverse of a Matrix by Cofactor 22:42
- Inverse of a Matrix by Cofactor and Example
- More Example
- List of Non-Singular Equivalences 43:07
- List of Non-Singular Equivalences
- Example
- Cramer's Rule 52:22
- Introduction to Cramer's Rule and Example

### Linear Algebra Online Course

### Transcription: Cofactor Expansions

*Welcome back to educator.com, thank you for joining us, this is linear algebra, and today we are going to continue by discussing co-factor expansions and using the co-factor expansion of a matrix to compute determinants, and to also compute the inverse of a matrix.*0000

*Now we did talk about the inverse of a matrix last time, and although this particular procedure that we are going to show you today with co-factor expansions with something that we define in a minute called the ad joint.*0018

*It is the radically, useful computationally it's not something that you want to use for more than a 3 by 3 or a 4 by 4, but again for theoretical reasons, it's always nice to be introduced to it, and to see how it functions.*0026

*Let's just dive in and write now...*0038

*Okay, let's start with a definition...*0043

*... Okay, that's always we will let A ...*0051

*... Be an N by N matrix...*0059

*... And we will let M, capital M _{ij} be the N - 1 times N - 1.*0063

*Sub matrix, and again these definitions are more for formal purposes once we actually do examples, anything that seems a little strange and unusual here will make a lot more sense...*0082

*.. Of A obtained...*0093

*... By deleting, the ith row, and jth column...*0102

*Now let's say so let A be an N by N matrix, let M _{ij} be the N - 1 by N -1 sub matrix of A obtained by deleting the ith row and the jth column.*0121

*Now the determinant of this M _{ij}...*0133

*... Is called...*0143

*... The minor...*0147

*... Of the entry...*0151

*...A _{ij}, so for example if we had A_{32}, that would be the third column, second entry, we would knock out that row and that column.*0158

*We would take the determinant of what was left over, and then that's called the minor of that particular entry, and again we will do an example and it will make more sense.*0169

*One mo0re definition, the co-factor...*0178

*... Of A _{i}, oops let me put an i there of ij, which we denote...*0188

*... Capital A _{ij}, so co-factor, we use the capital, minor we use the M...*0201

*... Is the following A _{ij} equals -1, raised to the power of I + J times the determinent of M_{ij}.*0211

*Okay, don't let all these subscripts and i's and j's and -1's scare you, let's do an example and it will make a lot of sense, so let's define our matrix A over here.*0228

*We will have (3, -1 and 2), we will have (4, 5, 6), and the third row will be oops, excuse me...*0241

*... We will do (7, 1, 2), now M _{12}, so M_{12}, this is the sub matrix we get from crossing out the first row, second column.*0252

*If I knock out first row second column, I am left with (4, 6, 7 and 2), so we have (4, 6, 7 and 2).*0268

*And that's exactly what it is, first row second column, go back to the original matrix, cross out, and the numbers that you have left over, those form sub-matrix.*0282

*In this case 2 by 2, because the original was 3 by 3, so we cut and remember up the definition N-1, by N - 1, 3 by 3 becomes a 2 by 2.*0292

*Now if we take the determinant...*0302

*... Of this M _{12} and again the determinant of a 2 by 2 matrix is just going to be, put some parenthesis, it's just that times that - that times that, so 2 times 4 is 8 - (-42).*0307

*What we end up with is -34, so let's put that aside for a second, so we have the minor, we have the determinant of the minor, and then we have that other thing that we defined which is the co-factor.*0322

*Well the co-factor of (1, 2) is -1, and the power that you raise it to is the sum of this (1, 2) right here, so it's 1 + 2 times the determinant that we got.*0337

*M _{12}, now -1 to the third power is -1, so it's -1 times the determinant we found already, which is -4, so our co-factor is 34.*0354

*Lets go through this again, we have a matrix, we have a minor, we have the determinant of that minor, and we have something called the co-factor.*0371

*Our matrix, this is our original matrix right here, let me actually use, so this is our original matrix, we decided to take the minor, the M _{12}, which means crossed out the first row, second column, so we crossed out the first row , the second column.*0380

*What we are left with was A 2 by 2, that's our minor, it is a matrix.*0395

*When we take the determinant of that minor, we actually end up getting this number right here, so this is a number when you take a determinant , remember a determinant actually gives you back a number.*0400

*And then what we do is we find the co-factor, the co-factor (1, 2) is the determinant that we got, multiplied by -1, raised to the power of the sum of the indices, ! + 2.*0410

*Let's do another example, go back to my, actually let me go back to blue ink here, so let's see this time let's do, let's calculate the M _{23} minor.*0428

*When we go back to our original matrix and we knock out the second row, and third column, what we end up with is another, the 2 by 2, which is (3, -1, 7 and 1).*0444

*When we take the determinant of M _{23}, it is going to be 3 times 1 - (-1) times 7, it's going to equal 3 - (-7), which is 3 + 7, we get 10.*0461

*That's our determinant, and now our co-factor A _{23}, that's equal to -1, raised to the power of 2 + 3, the row + the column, times our determinant.*0478

*I will just go ahead and put that here, okay -1 raised to the fifth power is -1, so you end up with -10.*0494

*Once again, we have our matrix, the original matrix, we knock out the second row and third column, because we are interested in the M _{23}.*0505

*Anf we have a matrix, we take the determinant of that matrix, an then from that we derive something called the co-factor, so it's the co-factor that's actually going to be the real important thing that we continue to deal with.*0514

*Okay, so now what we have this thing called a co-factor, as it turns out we can use it to evaluate determinants, so before what we do this, we use the properties of the determinants to manipulate the matrix.*0528

*Find as many as 0's as we can, may be factor out some numbers, simplify things as much as possible, essentially put it into upper triangular form if you remember from the last lesson.*0541

*And then just multiply everything along the main diagonal, well this is another method of actually doing it, and again computationally it may not be as efficient, it may or it may not, it depends on the situation.*0551

*But theoretically it comes in very handy, and it will make more sense as we proceed with linear algebra, but for right now let's just go ahead and work on actually finding a determinant using this co-factor expansion....*0563

*Okay, we have a theorem, okay, A = A _{ij} and remember this symbol is just a short hand symbol for the entire matrix, all of the entries, okay.*0579

*A is N by N, okay...*0597

*... Then for each I, which is less than or equal to N, greater than or equal to 1...*0604

*The determinant of the matrix A is actually A _{i1} times the co-factor.*0615

*I1 + A _{i2} times the co-factor, i2 + so on as many rows or columns A_{in}, oops, that should be a small A.*0624

*All this...*0645

*... _{in} times A_{in}, now there is another one of these four of the columns and but again we have a lot of indices, we have a lot of A's, you know lowercase, uppercase.*0649

*Instead of sort of throwing at a bunch of symbolism, let's just go ahead and do an example, and it will make more sense, essentially what this says is that I can pick a particular row or column of my choice, and I can expand that matrix along that row or column.*0663

*We will just see what we meant in a minute, that's all this is saying, that you can sort of add that entry of the row and, let's say you pick the first row, you can take the first row entry times the co-factor for that entry.*0679

*The second entry in the row and the co-factor for that entry, add them all together, and as it turns out, you end up getting the value of the determinants.*0692

*Let's just jump into the example.*0701

*We will let A = (1, 2, -3 and 4) (-4, 2, 1, 3) (3, 0, 0, -3), (2, 0, -2 and 3).*0704

*This is our matrix, now we take a look at this matrix and we want to make things as easy as possible for us, so we want to pick the row or the column that has the most number of 0's, because that way those terms just drop out of this sum up here.*0728

*They actually don't show up at all, so it makes, makes our life a lot easier, so when I look at this, I see the third row, has two 0's.*0744

*I am going to go ahead and expand along this row, now I am going to give you a little bit of sum as little bit of check board pattern.*0753

*In this case we have a 4 by 4 determinants, and as it turns out, remember that -1 raised to the power of the I + J, well as it turns out when you, I + J is going to iterate, 1, 2, 3, 4, 5, 6, 7.*0764

*And what happens is that -1 becomes -1 + 1 - 1 + 1, so instead of keeping track of all the symbolism, as far as the definitions are concerned, and this is going to draw out the +, - pattern for a 4 by 4, and also a 3 by 3 for you so that you know.*0779

*+, -, +, -, basically it's just alternating +'s and - all the way through, -, +, and you can never have a + next to a - vertically or horizontally.*0798

*that's all it is, so you can do this for a 5 by 5, 6 by 6 if you need to free yourself, that's +, that's -, that's -, +, -, -, +, -, +.*0809

*And here we have +, -, +, and you will see what this means in just a minute, -, +, -, +, -, +, so these are the things that you want to keep in mind when you do your co-factor expansion, okay.*0820

*Now we have decided to actually expand this along the third row, so here is what the expansion look like, now I take, I am doing 1, 2, 3, 4, so I am going to have four terms.*0834

*Well my first term over here is a 3, and if I go over here and if I take a look, I see that this is a positive, so that means that whatever term is there, in the end I stick a positive sign in front of it.*0849

*In this case it's going to be a positive and I like to actually put my positives in front of my positive values, it's just to have it that I have, it's up to you, negatives of ‘course, you need positives, you don't necessarily need, but for me it helps keeps things consistent and balanced.*0861

*3, now we said that you are going to expand it along this and the co-factor is if you erase that row and erase that column, what you are left with is this, this,. this number, this, this, this number and this, this, this number, a 3 by 3.*0876

*Lets go ahead and...*0894

*... (2, -3, 4), (2, 1, 3), (0, -2 and 3) okay, now we move to the next one, I'll go ahead and write...*0898

*... What I actually know, I won't write up, because this is 0, it's just going to be a -...*0913

*... 0, this is 0, so it's going to be a + 0, now our last term -, minus is here we put that there.*0921

*This is -3...*0935

*... Now we knock out that row and that column, but that's in and we are left (1, 2, -3)...*0940

*... - (4, 2, 1)...*0953

*... (2, 0, -2), so now what we have to evaluate is this thing right here, okay.*0965

*We have done a co-factor expansion, we chose our third row, because it has a couple of 0's in it, we took this entry times...*0976

*... The co-factor of that, we took, if we were to actually write out this one, we would knock out that entry, and then that entry, and here this one we took this entry times its co-factor.*0987

*Now we have this, so now we have a 3 times the 3 by 3, couple of 0's, and a -(-3) times another 3 by 3, so now let's evaluate this, evaluate this and we will put everything together.*0999

*Go back to my black ink here, I am going to rewrite what it is that we had, so we had 3 times (2, -3, 4), (2, 1, 3, 0), -2 and 3.*1013

*We have -0, we have +0, we have -(-3) times (1, 2, -3).*1030

*Again with linear algebra, there is a lot of minus's, plus's and numbers floating around, I'd like to write everything, I don't like to do it in my head, and for example I don't turn this into a +3, as I am doing it.*1041

*I wait until the end, so that I make sure that as I am going down the list, every symbol that I have is completely explicit and clear, not something that I have forgotten (-4, 2, 1, 2, 0, -2).*1054

*Okay, so let's go ahead and expand this one, now this one, we can probably do, we can expand it aong this column or this row, because it has a 0 in it, it doesn't really matter which one.*1070

*Let's go, you know what I think I am going to go ahead and take this row, red, I think I am going to go ahead and expand along that row, so I have got...*1083

*... The 0 is gone, so I have a -2, now I go back to my pattern, let me rewrite my pattern here, so that I remember it for a 3 by 3, +, -, +, -, +, - and +, -, +.*1100

*Here it's going to end up being...*1115

*... -2 times, and when I knock out that row and that column, I get (2, 4, 2, 3).*1122

*And then I have this entry right here, the 3, well that's a + sign, so i put a + sign, I put the 3.*1136

*And now I illuminate the row and the column what that belongs to and I am left with (2, -3), (2, 1).*1147

*Okay, that is going to end up equaling...*1156

*... 2 times 3 is 6, 6 - 4 times 2, that's -2, well okay so this determinant is -2, -2 times -2 is 4 times -1 is -4.*1163

*That's why I want to keep track at everything, and I don't want to do it in my head early on, I want to see every negative sign and every positive sign.*1180

*Here we have 2 times 1 is 2, - of -3 times 2, so - of -6 is 8, 3 times 8 is 24, positive, 2, so this equals 20.*1187

*Okay, so this thing right here, this determinant is equal to 20, now we will do this one, okay that will be our, let's call this, so we will call this number 1, that's our first determinant number 1.*1204

*And we will call this number 2, so now we are going to do the second determinant, now we take a look in same thing, we have a 0, 1 here, so let's go ahead and expand it along this, so we look down here, it's positive.*1221

*It's going to be 2 times, well we eliminate the row and the column what that belongs to, (2, -3, 2 1), (2, -3, 2, 1), the 0 doesn't matter, but I'll go ahead and put it in anyway.*1235

*Actually you know what, we don't need it, it's not that important, and the -2 over here, it is positive, so...*1247

*... -2, I would like to write it that way, and when I eliminate that row and that column, I am left with (1, 2, -4, 2), (1, 2, -4...*1257

*... -4 , 2 okay, and when I do this multiplication, I end up with 16 - 20 = -4.*1275

*Now this is, these are just the square determinant, notice I haven't taken care of the 3 and the - (-3), so now I am going to do that, so I have my final answer as 3...*1285

*... Times the 20, okay, now it's...*1297

*... - (-3) times -4.*1304

*And when you do all of that you end up with 48...*1313

*... Again this last part, let's do it again, I got 20 for this determinants, I have got -4 for this determinant, but I have the 3 and the - (-3) from the 4.*1319

*I brought those down, 3 times the 20, - sign is here, -3 is here and the determinant that we solved for this one is -4.*1331

*Make sure you keep track of all these, a single - sign will make you go in an entirely different direction, that's why it's really important to write everything out and do it this way., it is the best way to do it.*1341

*B is clear and explicit is possible, don't do anything in your head along the way, okay.*1352

*Let’s see what else we have here...*1361

*... Let's talk about the inverse of a matrix by co-factor, so what we just did was introduce co-factors, we did a solved determinants by co-factors, and now we are going to see if there is a way to actually find an inverse by co factors, and there is.*1365

*So far the way that we have been doing inverses is actually very wonderful computational method, so if I have some matrix A, which is N by N, I actually end up forming the augmented matrix if you remember.*1380

*Okay, let's just put that there, I ended up forming this matrix, and then I subjected this matrix to Gauss Jordan elimination, in order to end up with the reduced row echelon form, and if it actually has a matrix.*1396

*If it has an inverse, that A turns into the identity matrix, and the identity matrix turns into some matrix B, well as it turns out, that B...*1410

*... Is the inverse, wonderful procedure to do it, always works, if it doesn't turn out to be an identity matrix, in other words if you get a row of 0's or a row of inconsistence.*1423

*Then that just means that the identity matrix, that means that the inverse doesn't exist, excuse me, we are drawing around all kinds of terms, identity matrix, inverse, determinants, try to keep them all straight, I will try to keep them all straight, okay.*1433

*Now let's introduce another way using co-factors, start with the definition...*1449

*... Okay, we will let A = be the matrix, IJ, okay and it is N by N of ‘course.*1457

*Well let's just write it out, N by N, then...*1470

*... Then N by N matrix, okay...*1478

*... ADJ of A, of the, called the ad joint...*1486

*... Of A...*1500

*...Is the matrix...*1507

*... Who is I, Jth entry.. is a co-factor...*1512

*... JI, in symbols, it looks like this, ad joint of A and sometimes you will see parenthesis around, sometimes not.*1523

*A _{11}, A_{21}, A_{n1}, A_{12}, A_{22}, A_{1n}, all the way down to...*1538

*... A _{nn}, okay, very careful here.*1559

*Let's go through this, very carefully, the first thing that I would like you to notice, so we have, if we start with some matrix, and we want to form the ad joint of that matrix, okay, what we do is...*1566

*Let's say for example the...*1569

*... For that entry, for the first row, first column entry of the ad joint, we actually take the co-factor of the entry for the original matrix.*1582

*But what's interesting is notice that the entry for the second, for the first row, second column, is the adjoint for the second row first column.*1596

*Everything is sort, is transposed, so what you end up is that, the order is reversed here, so when you are going along a matrix, and you are going along the rows and columns of that matrix.*1609

*When you form the ad joint, you are actually going to reverse the order, so we will give you an actual procedure, the method for doing this properly, so that you will always end up getting it right, okay....*1622

*... Best method...*1639

*... For forming the ad joint of A, very simple 1...*1647

*... Form, the co-factor matrix of A and then another words, the matrix of the co-factors of A, and again it will make sense when we do an example.*1657

*And then just take the transpose, that’s the best way to do it, take the transpose instead of trying to do each entry and try to remember where it goes.*1673

*Just do it for the matrix straight, and then just flip it along the main diagonal, okay let's do an example.*1682

*We have our standard matrix A, it is (3, -2, 1, 5, 6, 2, 1, 0, -3) okay.*1690

*And just to remind ourselves, I will go ahead and put +, -, +, -, +, -, +, -, +...*1706

*... Our pattern, okay, so now we are going to actually find the co-factor for this entry, this entry, this entry, for all of the entries, and then we are going to put them in into a new matrix, those entries, and then we are going to flip it.*1716

*Okay, so A _{11}...*1730

*... = well, remember the co-factor, + then when you knock out this row and that column, you are left with (6, 2, 0, -3)...*1736

*(6, 2, 0, -3), when I take the determinant of that, it's 6 times -3, which is -18 - 0 +, so it's -18, that's that entry.*1747

*Now we will go to, where should I do it, I will do it, I will do it over here, okay, I know, it's okay I will do it over here.*1762

*A _{12}, okay so this is the co-factor for the entry, first row...*1778

*... Second column, so it's a - sign, right, so - and then if I knock out this row, this column, first row second column, I get (5, 2, 1, -3) left over.*1786

*(5, 2, 1, -3) left over, 5 times -3, -15, -2 times 1 -17, - (-17), I end up with +17.*1802

*Again arithmetic is a real issue here, let me erase some of this here...*1817

*Okay, I want to actually be able to do it right underneath, so A _{13}, so we are so on the first row third column, that entry is that, okay.*1825

*It's positive, when I knock out, the row and the column that, that entry is now left with (5, 6, 1, 0).*1837

*Okay, and the, 5 times 0 is 0, -6 times 1 + -6...*1849

*Okay, now let's move to the second row, so now we want to find A _{21}...*1858

* _{21}, well negative sign, when we come over here, the entry is a 5, we knock out that row, that column, we are left with (-2, 1, 0, -3).*1866

*That determinant is equal to -6...*1886

*We want to do A _{22}...*1893

*...Positive, we are here, when we knock out that row, that row , that column, we are left with (3, 1, 1, -3)...*1897

*... (3, 1, 1, -3), this is -9 - that, you get a -10, okay.*1909

*Let's go to A _{23}...*1921

*... A _{23}, actually you know what; I am going to, so that we actually have something to refer to.*1927

*I am going to rewrite the original matrix here, and yeah, that's not a problem, so A, our original matrix was (3, -2, 1), (5, 6, 2), (1, 0, -3).*1937

*We want to be able to refer to it, and we said that the last thing that we are working on was the co-factor for the second row, third column.*1954

*Okay, so we go to second row third column, and row third column here, it's A negative, +, -, +, -, +, -, and then when we knock that out, and that out, we get (3, -2, 1, 0).*1966

*(3, -2, 1, 0), 3 times 0 is 0, -(-2) is + 2, -(-2), okay now we will work on the third row A _{31}, positive, negative, positive, so it's positive, 3321 When I knock out that row, that column, I am left with (-2, 1, 6, 2), (-2, 1, 6, 2) when I solve that I get -10.*1980

*I have a (3, 2), so positive, negative, positive, positive, negative, negative, and I knock out that row, that column I am left with (3, 1, 5, 2).*2014

*Okay, I end up with -1, and my last one, the co-factor for the third row, third column, +, -, +, +, -, +, is +.*2028

*When I knock out that column, that row, I am left with (3, -2, 5, 6), (3, -2, 5, 6), and I should end up with 28 if I had done my arithmetic right, okay so now we can put all of these numbers into a matrix.*2044

*We can form...*2064

*... We start with, let's call it the, well we don't have to call it anything, the co-factor...*2067

*... Matrix, of A, okay that's our first step, we form the co-factor, then we take the transpose, so we put all of the numbers that we got in (-18, 17, -6).*2078

* (-6, -10, -2, -10, -1), so this one, the last one we did, and 28, and now after we do this, we want to take the transpose of this, okay.*2095

*let's go ahead and take the transpose, and we end up with the final matrix, which looks like this, so it's going to be -18, -6, -10, 17, -10, -1...*2115

*-6, -2 and 28, so I started with a matrix, I found the co-factors for each entry, in which numbers essentially.*2137

*And I take those numbers and I put them into, and I form a new matrix, okay, this is my co-factor matrix, and then that matrix that i got, I take the transpose of that.*2152

*And I end up with...*2163

*... This is my ad joint...*2168

*... It's a bit of a process, but this is how you do it, okay, that's the, here, now let's use our ad joint to come up with, the actual inverse.*2174

*We see, we will let A, be n by N...*2192

*... Then...*2203

*... Yeah let me write out the theorem and then we will settle with it, A times the ad joint of A...*2208

*... Okay, equals the ad joint of A, times A is equal to the determinant of A times the identity matrix, and the only reason the identity matrix shows up here.*2217

*Is that here we have two matrices multiplied by each other, that gives you matrix, remember the determinant is a number, so and fairer, i want to take turn a number into a matrix, and multiply it by the identity matrix.*2231

*It just means multiplying everything on the main diagonal, so it's a way it is converting a number to a matrix, that's what the only reason that it shows up here.*2241

*this theorem says that if I take the original matrix, and I multiply by its ad joint, or if I do it the other way, if I take the ad joint and multiply by the original matrix, they are both defined, because it's just N by N either way.*2250

*I end up actually with the determinant times the identity matrix, now again since this is an equality, anything that I do to one side, I can do to the other and maintain the equality, so let's just fit with this a little bit.*2262

*Let's take... let's take this one and let's take that one, we can, I just take randomly in order, so let's just go ahead and knock this out, the equality is still retained, but it doesn't really matter.*2276

*I am going to multiply both of these by inverse ion the left.*2290

*When I do that, the equality is retained, so the A inverse, times A times ad joint of A, equals A inverse times determinant of A times the identity matrix, okay.*2298

*Now, well A inverse times A , and remember associativity, A inverse times A is just the identity matrix, so I end up with the identity matrix times the ad joint of A is equal to...*2309

*... A inverse times the determinant of A times the identity matrix, and again the identity matrix as far as the matrices are concerned.*2329

*It is the identity, in other words, just acts like a 1, so when we do 5 times 1, we don't necessarily put that 1 there, we just say its 5, so, for all practical purposes, we can ignore these, and now since determinant is a number, I can divide both sides by that number.*2341

*Remember I can't divide by a matrix, but I can divide by a number, the determinant of A, so we can ignore this and this, they are just, they are there.*2360

*Now, the determinant, those cancel and I am left with the A inverse is equal to the ad joint of A, divided by the determinant of A.*2371

*This is my formula, so now if I am given a matrix and if I want to find the inverse, I can do it one of two ways, I can go ahead and set up that augmented matrix, and convert the matrix to the identity if it is such and the identity matrix will turn into the inverse if it exists.*2383

*If not then, the inverse doesn't exist, and here, or I can find the ad joint matrix, which again I can do very simply with mathematical software, it's not a problem.*2400

*And I can divide it, oops, I forgot to put the A, I can divide it by a determinant, well notice something really interesting here that determinant is a number.*2409

*Well, if the determinant is 0, we know that division by 0 is not defined, so in this particular case, if you know the determinant of a matrix is 0, that automatically tells you that the inverse doesn't exist., so now we have, we are going to actually write that as a theorem.*2420

*Okay, A is non-singular, and we will remember non-singular meant invertible, it has an inverse...*2440

*... Well, if and only if, which means is equivalent to the determinant of A, not being 0, excuse me, so if I am given a matrix, and if I take the determinant of that matrix.*2455

*Again with math software, however i want to do it, if the determinant ends up not being 0, I know the matrix, I know the inverse exists.*2469

*If the determinant is 0, this formula that establishes a relationship between the inverse of determinant of the ad joint tells me I can't divide by 0 that tells me that the inverse doesn't exist.*2476

*Okay...*2490

*Now, in the beginning of the lesson I mentioned that computationally this is probably not the best way to go about finding an inverse, but it is kind of interesting from a theoretical point of view to see that if you start with some inverse, and if you end up sort of fiddling with the numbers, the square array of numbers.*2495

*And you end up with something called the ad joint, or if you divide by it's determinant, it ends up being related to its inverse, this is what we look for in mathematics.*2511

*There is no reason in the world to believe that this is true, there is nothing in intuitions to lead you to actually investigate that this would be true, but when you start fiddling with things, and when you start sort of following logical conclusions and seeing where particular mathematical derivations lead you.*2519

*You end up with something really extraordinarily beautiful, the ad joint of matrix, its inverse, it's ad joint, and it's determinant are actually related.*2536

*that's very strange, and again there is no reason it should be that way, but there it is, we will lose dated a fundamental fact about nature, a fundamental fact not about mathematics, but about how numbers behave, how collections of numbers behave in this case.*2546

*Okay, let's go ahead and add to our list of no-singular equivalences, you remember last time we have a list where we said if A is non-singular meaning invertible, we can draw other conclusions that are equivalent to it.*2562

*like we said before we are going to continue to add to that list, and the list is going to get rather long, but it's going to be very powerful list...*2574

*... Okay, so we have, A is non-singular invertible, okay...*2587

*... that's the same as saying that if I take A and multiply by some variable matrix that the system, the linear system, the homogeneous system AX = 0, has only the trivial solution.*2597

*Okay, so if its invertible, if it has a matrix, the AX = 0, the homogeneous solution, so just by knowing that something is actually invertible, taking a determinant and realizing that the determinant is not 0.*2609

*I can tell you something about the homogeneous solution, I have made a qualitative statement about it, I don't have to worry about trying to finding it, because it tells me it's only trivial, it's not worth finding.*2623

*A is row equivalent to the identity matrix meaning, with those manipulations of exchanging rows, exchanging columns, multiplying one by the other, aI can convert it to the identity matrix.*2633

*AX = B, has a unique solutions of every b, so if I have a matrix, an N by N matrix A, and I know that the inverse, I know that the determinant is not 0.*2644

*that means the inverse exists, I know that the, if I have, that the non-homogeneous solution, that there is actually an unique solution, not an infinite number of them, and not no solution.*2655

*And of ‘course the last one, which is the one we did today, if it is non-singular invertible. that's the same as determinant not being equal to 0.*2667

*Okay, let's do an example, okay we want to compute the inverse of the following matrix if it exists.*2677

*Let's try, let's try blue ink, how's that?, so we have our matrix (4, 2, 2), (0, 1, 2), (1, 0, 3), so we have a 3 by 3 matrix.*2688

*First thing we want to do is find it's determinant, okay, so the determinant of A, I am going to do a co-factor expansion, and I think I am going to go expand it along this column.*2705

*I could do this this row, because a 0 in it, but I am going to go ahead and expand along this column this column, so again let me over here, write +, -, +, -, +, -, +, -, +.*2722

*When I go according to this column, the first term is going to be a +, second is going to be a -, which it doesn't matter it's a 0, and it's going to be a + over here.*2733

*The determinant is equal to 4 times; I knock out 1, 2, 0, 3...*2742

*... - 0 + 1 times (2, 2, 1, 2).*2751

*1 times is 3, -0, 4 times 3 is 12...*2761

*... And this one of ‘course doesn't matter, 2 times 2 is 4, -2 is 2 times 2 so it is +2.*2768

*Our determinant is 14, it’s not equal to 0, so that the inverse exist, now let's actually find it using our formula, okay.*2779

*We want to form, our first step is we want to form the matrix, the co-factor of A, we want to take A, which let me rewrite it again because just to make a little more clear what matrix we are dealing with.*2791

*(4, 2, 2), (0, 1, 2), (1, 0, 3), if we want to form the co-factor, then we want to take the transpose of that okay, so let's go ahead and do our co-factors.*2804

*A _{11}, so i go up here, A_{11}, that means I knock out this group, this row, this column and I am left with (1, 3, 2, 0).*2819

*The determinant of that is 1 times 3, is 3 - 0 is 3, its positive, A _{11} is 3...*2831

*... A _{12}, I go up here first row second column, I knock that out, I knock that out, I am left with (0, 2, 1, 3), okay.*2843

*And since this is going to be a negative, 0 - of -, this is -2 is the determinant, but I have stick a negative sign in front of it, so it ends up being +2.*2855

*A _{13}, I knock out that column, that row, I am left with (0, 0, 1, 1), so it''s 0 -1 is -1, and it's positive, so that's equal to -1.*2870

*A _{21}, I will do one more, and then I will just write down what the others are, so now we are on the A_{21}, which is second row first column.*2887

*I knock out this column, knock out that row, I am left with (2, 2, 0, 3), 2 times 3 is 6 - 2 times 0 is 6, so 6 - 0 is 6.*2898

*However we are expanding, we are using this co-factor, which is a negative, so it's -6.*2911

*And then when we continue along this fashion, we get A _{22} will go +10, we get A_{23} is equal to +2.*2919

*A _{31} is equal to a +2, A_{32} is equal to a -8, and A_{33}, I'll actually do this one, 33 we are down here.*2932

*Knock out this one, knock out that one, I am left with (4, 2, 0, 1), 4 times 1 is 4, -0 is 4.*2947

*Positive, positive 4, okay, so now I am going to actually put this, I did like this, because these are the numbers that you are going to arrange, your matrix is actually going to be (3, 2, -1, -6, 10, +2, 2, -8 and 4) a co-factor matrix.*2958

*Lets form that, so our co-factor of A is equal to (3, 2, -1, -6, 10, 2) and we have (2, -8, and 4).*2978

*Now we want to subject that to transposition, and what we end up with is just flip the rows and the columns.*2998

*(3, -6, 2, 2, 10, -8, -1, 2 and 4).*3007

*This is our ad joint...*3019

*... Okay and now we said that our determinant of the original matrix is equal to 14, we know that the A inverse oops, well that was interesting...*3023

*... We know that A inverse is equal to the ad joint of A, divided by the determinant of A, well we just take this ad joint which we just got.*3046

*Divide every entry in there by 14, so we get three 14's - 6 14's, 2 14's.*3061

*Now I wanted to talk to you about reduction, yes, 2 14's is 1 se7th, 6 14's is 3 7th, it's up to you, i personally, I don't like, I like to leave my numbers the way that I found them.*3072

*Reducing is fine, if you want to, if you don't want to it's perfectly acceptable too, I like the degree of consistency, i don't like 7's and 14's floating around in my numbers.*3086

*It just helps me, thus so i actually don't reduce my numbers, I know most high school math's teachers will probably kill me for that, but there it is.*3093

*Okay, so two 14's, ten 14's - eight 14's - 1 over 14, 2 over 14 and 4 over 14.*3103

*This final matrix is our inverse that we were looking for, again it the ad joint of A over the determinant of A, determinant exists, we are good.*3119

*If the determinant is 0, we don't have to bother trying to find an inverse, so very, nice theorem to be able to use that.*3135

*Okay, we are going to close off this particular lesson with a really beautiful theorem called Cramer's rule, you probably remember it from your high school course, i, perhaps you didn't see it.*3142

*I imagine most of you have probably seen it though, it offers a way to find a solution to a linear system that is the same number of unknowns as variables, same number of unknowns as equations.*3154

*Let us write out, let me go back to my black ink actually, let...*3170

*A _{11}X_{1} + A_{12}X_{2} + A_{1n}X_{n}.*3178

*A _{21}X_{1} + A_{22}X2_{2} + A_{2n}X_{n} and then we will work all way down, A_{n1}X_{1} + all the way down to A_{nn}X_{n}.*3190

*This is a system of, oh let me put my solutions here, B _{1}, B_{2}, all the way to B_{n}, so this is a linear system.*3212

*N equations and unknowns, M by N system okay, the theorem says if the determinant of A and the determinant of A is just the coefficient matrix, in other words I take all of these coefficients...*3225

*... And then I put them in matrix form...*3245

*... If the determinant of A does not equals 0, then...*3249

*... The system has the unique solution...*3259

*... X _{1} is equal to determinant A_{1} over determinant of A, X_{2} = determinant of A_{2} over the determinant of A etc.*3273

*X _{3}, X_{4}, okay and let's finish this off, we are...*3293

*... A _{n} is obtained...*3306

*... From A...*3314

*... By replacing the N'th column...*3319

*... Of A, with the vector B, which is the solutions, and again it will make sense when we actually do an example here.*3330

*We have the following system -2X _{1} + 3X_{2} -X_{3} = 1.*3341

*X _{1} + 2X_{2} - X_{3} = 4.*3356

*And -2X _{1} - X_{2} + X_{3} = -3, okay.*3365

*Let's do our matrix A, it is just the coefficients, let me do it in red here, so we have (-2, 3, -1, 1, 2, -1)....*3375

*... (-2, -1 and +1) good, and let's do our B, vector m, which is just spread here, (1, 4, 3) okay.*3392

*My X _{1} is going to equal, now when I take the determinant of this, let's actually go ahead and write out what we have.*3407

*I am not going to go through the process of it, I use mathematical software and the determinant is -2, okay, so now when I take A, let me erase this here...*3416

*... A _{1}, I get A_{1}, by replacing this first column, 1, with that number, so(1, 4, 3) with this whole vector I mean.*3434

*3 and then (3, 2, -1, -1, -1, 1) okay, when I take the determinant of that I end up with -4.*3447

*therefore X _{1} is equal to the determinant of A_{1}, which is -4, over the determinant of A - 2 is 2.*3461

*This is my final answer for X, A _{2}, well it's equal to this with that second column replaced by that vector, so I end up with (-2, 1, -2, 1, 4, 3, -1, -1, 1).*3472

*I subject that to determinant, mathematical software, I end up with -6, therefore X _{2} = -6 over -2, which is equal to 3.*3493

*And my final A _{3} is equal to (-2, 1, -2, 3, 2, -1) and the third column is replaced with that.*3507

*(1, 4, 3)...*3522

*... I subject that to the determinant function and I end up with -8, therefore X _{3} = -8 over -2, my final answer is 4, X_{1} is 2, X_{2} is 3, X_{3} is 4.*3528

*For my coefficient matrix, and I take the determinant of the A _{1}, A_{2}, A_{3} matrix that I obtain by sticking the solution vector in the particular column for the, for X_{1}, X_{2}. X_{3} and I solve it that way.*3547

*Thank you for joining us at educator.com, we will see you next time.*3568

1 answer

Last reply by: Professor Hovasapian

Wed Oct 30, 2013 3:11 AM

Post by Fawzy Hegab on October 30, 2013

I want to thank you professor for this great Lectures!

2 answers

Last reply by: Professor Hovasapian

Sun Sep 1, 2013 10:30 PM

Post by Christian Fischer on September 1, 2013

Hi Raffi,

Great video! Just a quick question: You state that in Cramers Rule A1 is formed by taking the n'th column of A and replacing it by vector b, but to me at 57 minutes it looks like you replace column 1 (and not "n" which i consider to be the last column in the matrix) with vector b. Is that an error, or have i misunderstood that part?

Best wishes,

Christian

1 answer

Last reply by: Professor Hovasapian

Mon Jun 10, 2013 3:18 PM

Post by Manfred Berger on June 7, 2013

I know I'm jumping ahead a good bit, but I'll ask anyway. Are you going to come back and clarify what the adjont means in terms of finite dimensional linear operators? Most authors seem to be happy to skip that concept altogether.

1 answer

Last reply by: Jimmi Aastrom

Sat Jan 5, 2013 9:33 AM

Post by Jimmi Aastrom on January 5, 2013

Michael;

It probably doesn't matter at this point. But the problem is that Mr. Hovsapian wrote the last entry of the solutionvector b as 3 instead of -3.

If you calculate the determinants using 3 instead of -3, you get -10, -24 and -50 respectively, in stead of -4, -6, -8 which are indeed the correct values for the determinants.

1 answer

Last reply by: Professor Hovasapian

Sun Aug 19, 2012 8:55 PM

Post by michael bentt on August 19, 2012

Mr. Hovasapian:

Using Maple, I attempted to find the determinants you were seeking in "Cramer's Rule" section, part of the Cofactor Expansions topic, however, my values are different from yours.

I double checked and I derived the following values: det(A1) = -10, det(A2) = -24 & finally det(A3) = -50 whereas yours are -4,-6-8, respectively.

I've also included a screenshot of my work:

http://s1064.photobucket.com/albums/u365/mikebbc/?action=view¤t=06303fa4.png

Thanks in advance

3 answers

Last reply by: Fawzy Hegab

Wed Oct 30, 2013 2:47 AM

Post by Romin Abdolahzadi on October 10, 2011

His remarks are awesome! "There is no reason in the world to believe this is true. There is nothing in intuition to lead you to actually to investigate that this would be true. But when you start fiddling with things, when you start following logical conclusions, and seeing where particular logical mathematical derivations lead you, you end up with something really extraordinarily beautiful. The inverse, its adjoint and determinant is actually related! That's very very strange."