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Lecture Comments (6)

1 answer

Last reply by: Professor Selhorst-Jones
Thu Dec 4, 2014 5:25 PM

Post by Johnathon Kocher on December 3, 2014

Hello, I'm just dropping a comment to let you know the "Download Lecture Slides" and "Practice Questions" tabs are empty for this lecture.

1 answer

Last reply by: Professor Selhorst-Jones
Sun Mar 9, 2014 2:34 PM

Post by Christopher Hu on March 8, 2014

I tried the "alternative" method for a 4x4 matrix and it does not work. the "alternative" method only works for a 3x3, right?

1 answer

Last reply by: Professor Selhorst-Jones
Sat Aug 10, 2013 1:00 PM

Post by Ikze Cho on August 10, 2013

Does the "alternative method for 3x3 matrices" only apply for the 3x3 matrices or also for other matrices?

Determinants & Inverses of Matrices

  • The inverse of a matrix A is some matrix A−1 such that when we multiply them together we get the identity matrix, I. In other words, they "cancel" each other.
  • Not all matrices can be inverted. A matrix that can be is called invertible (or `nonsingular'). If it can not be inverted, it is singular. To be invertible, a matrix must have these properties:
    • The matrix must be square,
    • The determinant of the matrix must be nonzero.
  • The determinant is a real number associated with a square matrix. The determinant of a matrix A is denoted by either  det(A)  or |A|. [Although |A| may look similar to absolute value, it is the determinant of A and can produce any real number (including negative numbers).] If the determinant of a matrix is nonzero, the matrix is invertible, and vice-versa.

    (A) ≠ 0    
        A is invertible

    (A) = 0    
        A is not invertible
  • The determinant of a 2×2 matrix
    A =


    is given by

    (A) = |A| = ad − bc.
    A good mnemonic to remember this is to think in terms of diagonals: the down diagonal multiplied together then subtracted by the up diagonal (also multiplied).
  • To take the determinant of a larger matrix, we need the two following concepts.
    • Minors: For a square matrix A, the minor Mij of the entry aij is the determinant of the matrix obtained by deleting the ith row and jth column.
    • Cofactors: The cofactor is very closely based on the minor. It just multiplies the minor by 1 or −1 based on the location of the entry the minor comes from. The cofactor Cij of the entry aij is given by
      Cij = (−1)i+j Mij.
      We can also see this as an alternating sign pattern. It's very similar to a chessboard where one color is positive and the other is negative. See the video for a visual reference.
    The determinant of an n ×n matrix A is given by the sum of the entries in any row or column each multiplied by their respective cofactor.

    (A)  =  |A|
    ak1 Ck1 + ak2Ck2 + …+ akn Ckn
    a1k C1k + a2kC2k + …+ ank Cnk
    Note that this is true for any value of k (as long as 1 ≤ k ≤ n). That is, we can choose to do this process with any row or column and get the same result.
  • For a 2×2 matrix
    A =


    the inverse of A (if det(A) ≠ 0) is
    A−1 =1








  • For the most part, at the level of this course or any similar math class, you will probably not need to compute the inverse of a matrix any larger than 2 ×2. If for some reason you need to calculate the inverse of a matrix that is larger than 2×2 and you must do it by hand, see the bottom of these notes.
  • Given A and A−1, we have A−1 A B = B: they cancel each other out and have no net effect. This is because
    A−1 A  = I = A A−1.
    When multiplied together, they create the identity matrix I, which (as noted in the previous lesson) has no effect in multiplication.
  • It is important to note that if we multiply an equation by a matrix on both sides, we must choose a direction to multiply from and do the same for both sides of the equation. We must multiply on the left or on the right for both sides. [This is because PQ ≠ QP for most matrices.]
  • To find the inverse of larger matrices by hand, we will need some techniques we haven't learned just yet. In the first part of the next lesson, we discuss augmented matrices, row operations, and Gauss-Jordan elimination. If you're not familiar with these things, go check them out first. Here are the steps to find an inverse for any size matrix:

      1. For an n ×n matrix A, begin by creating an augmented matrix with the identity matrix In:



      2. Apply the method of Gauss-Jordan elimination (use row operations) to reduce A (the left side) to I. The result of the augmented matrix will be



      3. Finally, check your work. It's very easy to make a mistake in all that arithmetic, so check by showing either
      A−1 A = I       or        AA−1 = I.

Determinants & Inverses of Matrices

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

  • Intro 0:00
  • Introduction 0:06
  • Not All Matrices Are Invertible 1:30
    • What Must a Matrix Have to Be Invertible?
  • Determinant 2:32
    • The Determinant is a Real Number Associated With a Square Matrix
    • If the Determinant of a Matrix is Nonzero, the Matrix is Invertible
  • Determinant of a 2 x 2 Matrix 4:34
    • Think in Terms of Diagonals
  • Minors and Cofactors - Minors 6:24
    • Example
  • Minors and Cofactors - Cofactors 8:00
    • Cofactor is Closely Based on the Minor
    • Alternating Sign Pattern
  • Determinant of Larger Matrices 10:56
    • Example
  • Alternative Method for 3x3 Matrices 16:46
    • Not Recommended
  • Inverse of a 2 x 2 Matrix 19:02
  • Inverse of Larger Matrices 20:00
  • Using Inverse Matrices 21:06
    • When Multiplied Together, They Create the Identity Matrix
  • Example 1 23:45
  • Example 2 27:21
  • Example 3 32:49
  • Example 4 36:27
  • Finding the Inverse of Larger Matrices 41:59
  • General Inverse Method - Step 1 43:25
  • General Inverse Method - Step 2 43:27
    • General Inverse Method - Step 2, cont.
  • General Inverse Method - Step 3 45:15

Transcription: Determinants & Inverses of Matrices

Hi--welcome back to

Today, we are going to talk about determinants and the inverses of matrices.0002

Consider if we wanted to find x in the equation 5x = 10--pretty basic algebra, right?0006

We would cancel out the 5 by dividing it on both sides.0012

Or equivalently, we could think of this as multiplying by 5 inverse, which is just 1/5.0014

If we multiply by 1/5 on both sides, we cancel out the 5, because the multiplicative inverse to 5 is 1/5; that why it is 5-1.0020

What if we wanted to solve for the matrix X in the equation below?0028

We had some matrix A, times the matrix X, is equal to the matrix B.0031

We have this matrix equation; so we need to somehow cancel out A to get X alone.0036

It is the same basic idea; we just need to cancel out an entire matrix.0041

So, we need to multiply both sides by the inverse to A; this means we need to find the inverse to A.0044

If we can find this magical inverse, then we could multiply both sides.0050

We would have A-1AX and A-1B.0054

Well, the A-1 and the A will cancel each other out, and we would be left with X = A-1B.0058

So, we would be able to solve for that matrix, that unknown matrix, X, if we wanted to,0064

in terms of this A-1 and B, if we know what A and B are.0068

It is very similar to 5x = 10; we multiply by the multiplicative inverse of 5, 5-1, on both sides, to get what x is.0072

So, AX = B...we multiply by the multiplicative inverse of A on both sides to get that X alone.0079

Not all matrices are invertible; consider if we wanted to solve for X in the basic equation 0x = 0.0086

It would be impossible: the information about what x is has just been destroyed by that 0.0094

0 multiplied by anything is going to come out to be 0, so we don't have any idea what that x is anymore.0099

There is no way to cancel out 0, because 0-1 does not exist.0104

There are some special things out there that we can't invert.0109

There is no way to flip them to an inverse, because can't invert it.0112

You can't reverse the process of multiplying by 0; it is gone--the information is lost.0117

It is the same thing going for matrices: not all matrices can be inverted.0123

A matrix that can be is called invertible: if we can invert a matrix, we call it invertible, or we might call it non-singular.0127

If a matrix cannot be inverted, it is called singular.0134

To be invertible, a matrix must have two properties: the matrix must be square--it has to be a square matrix to invert;0138

and the determinant of the matrix must be non-zero.0145

So, what is a determinant? Let's start talking about determinants.0149

The determinant is a real number associated with a square matrix.0152

The determinant of a matrix A is denoted by either detA (like determinant of A--we are shortening it),0156

or vertical bars on either side of the matrix A.0163

Now, A may look similar to absolute value, but it is not; it is not absolute value--it is the determinant of A.0165

So, when it is vertical bars around a matrix, we are talking about determinant, not absolute value.0172

So, vertical bars around a matrix, unlike absolute value, can produce any real number, including negative numbers or 0 or positive numbers.0176

So, it is not limited to just giving out positive or 0, like absolute value; it is allowed to put out anything.0186

So, don't get confused by those vertical bars, thinking that that implies positiveness; it doesn't.0191

For the most part, though, I prefer this detA thing, this determinant of A; so that is the form that we will be seeing.0195

But occasionally, you will see it with the vertical bars, instead.0202

The determinant of a matrix has many important applications and properties.0204

There is a huge amount of stuff that this determinant is useful for.0207

But we are not going to get into that in this course.0211

In this course, we are only going to concern ourselves with one thing: whether or not a matrix is invertible, and the fact that a determinant tells us that.0213

If a determinant of a matrix is non-zero, then the matrix is invertible, and vice versa.0220

So, if the determinant of A is not equal to 0, then we know that A is invertible.0228

And if A is invertible, then we know that the determinant of A must not be equal to 0.0232

On the flip side, if the determinant of A is equal to 0, then we know that A is not invertible.0236

And if A is not invertible, we know that the determinant of A is equal to 0.0241

So, just remember that detA not equal to 0 means that it is invertible.0245

And that really works a lot like we are used to with the real numbers.0249

You can invert any number you want, except 0.0253

It is the same thing with matrices: you can invert any matrix you want, except for ones that have determinant 0.0256

All right, so think in terms of that: detA not 0 means invertible--you are allowed to invert; detA = 0--you are not allowed to invert.0261

So, let's see the determinant of a 2 x 2 matrix: if A = a, b, c, d, it is given by detA,0271

which is equal to this other way to write determinant of A, comes out to be ad - bc.0279

A good mnemonic to remember this is to think in terms of diagonals--0286

the down diagonal, ad, multiplied together, and then subtracted by this up diagonal here, cb or bc, so minus bc.0289

We subtract by that up diagonal.0303

Let's look at an example--let's do an example here: Multiply...0305

if we want to take the determinant of 5, 9, 3, 4, notice that we have these bars on either side.0309

If we have bars of some matrix inside, what that is saying is to take the determinant of that stuff on the inside.0315

Bars on either side is just like the bars on either side of the capital letter denoting the matrix.0326

It says to take the determinant of whatever is inside of there.0332

So, you will see that notation a lot; but when we are talking about just letters, I prefer that one.0334

OK, in either case, if we are taking the determinant of the matrix 5, 9, 3, 4--if we are taking this one right here,0339

the determinant of 5, 9, 3, 4, the first thing we do is take the down diagonal.0344

So, it is going to be 5 times 4; and then, it is going to be minus the up diagonal, 3 times 9, so 9 times 3.0351

5 times 4 - 9 times 3; we get 20 - 27, and that comes out to be -7.0361

Once again, the determinant can come out to be any number; it doesn't have to come out to a positive;0372

it just has to come out to any real number at all.0376

Minors and cofactors: first we are going to talk about minors.0380

Before we can look at determinants of larger matrices, we will need two concepts: minors and cofactors.0383

First, we are going to look at minors.0388

For a square matrix A, the minor, mi,j (remember, i is the row i; j is the column j)0391

of the entry ai,j is the determinant of the matrix obtained by deleting the ith row and jth column.0398

So, we go to this i,j location, this ai,j entry, and we delete out from that, vertically and horizontally.0405

So, we will take some location, and then we will delete out horizontally, delete out vertically, and group back together and see what is left.0414

For example, if we have a below, we would have m2,3; 2,3 means we are on the second row, and we are going to be on the third column.0421

We are looking at -8 as the epicenter of where this thing is.0431

That is the entry ai,j; so the entry 2,3 would be -8.0436

Now, we delete (the determinant of the matrix obtained by deleting) the ith row and jth column.0440

We delete this second row; we delete this third column; and we see what the matrix is that is left.0446

Well, the matrix that is left is 6, 2, -7, 3; that is all that hasn't been crossed out.0453

Now, we go and we take the determinant of that; we are taking the determinant with these bars,0461

because the minor is that you delete, and then you take the determinant.0466

So then, we just take 6 times 3, minus -7 times 2; 18 + 14...we get 32, so that is our minor.0469

Cofactor is very closely based on the minor.0480

The cofactor just multiplies the minor by 1 or -1, based on the location of the entry the minor comes from.0484

So, there is this shifting, flipping-back-and-forth pattern of positive/negative that is really deeply connected to the determinants of matrices.0491

The cofactor ci,j, the ith row, jth column cofactor, of the entry ai,j,0500

the entry in the ith row, jth column of our matrix A, is given by ci,j0505

is equal to -1 to the i + j times that minor i,j.0511

So, the -1 to the i + j is just a way of saying if it is going to be positive or negative.0517

-1 to the 0 is positive; -1 to the 1 is negative; -1 to the 2 is positive; -1 to the 3 is negative; -1 to the 4 is positive.0521

-1 to the even number is positive; -1 to the odd number is negative.0529

So, we can see this as an alternating sine pattern.0533

If we are in -1, to the row 1 + column 1, then that is going to be -1 squared; -1 squared comes out to be positive 1.0535

There we are at row 1, column 1.0546

If we were to instead, say, look at row 2, column 3, then it would be -1 to the 2 + 3, which is equal to -1 to the 5.0549

So, since it is to an odd number, it is going to be negative; so we get that negative there.0561

We can see this in terms of the i + j thing; but we can also see it in terms of this alternating sign pattern.0566

I would recommend, any time you are working with cofactors, that you just draw up the alternating sign factor to whatever size you are doing.0572

For example, if you are working with a 3 x 3, just draw out a 3 x 3 alternating sign pattern.0578

It always starts with a positive in the top left: so + - +, - + -, + - +.0584

And then, from there, you will be able to work from it and use that as a reference point; we will see that in the examples.0593

Thus, based on our previous example, when we took what m2,3 was (m2,3 was equal to0598

the determinant of 6, 2, -7, 3, because c2,3 is still going to be based around row 2, column 3;0605

so -8...cross out...cross out...6, 2, -7, 3...the same thing here; and then we just take the determinant of that);0612

but we are here in the 2,3 position in our alternating sign pattern (or alternately, if we want to look at it0621

in terms of -1 to the i + j--either way would end up working out the same); so we have this negative here,0629

so we have a negative showing up here, so that will end up coming out to -32,0636

because we already figured out that the determinant for that minor is 6, 2, -7, 3; that is what we get out of that.0641

And so, that came out to be positive 32; so when we have this sign on top of that, that is going to come out to be -32.0648

All right, so how do we actually take the determinant?0656

Let's apply this stuff: the determinant of an m x n matrix A is given by the sum of the entries in any row or column0657

(you can choose any row or any column at all), and you multiply each one of those entries0665

by the respective cofactor that would come out of that entry.0670

So, the determinant of A, which is equal to another way to say the determinant of A,0674

is equal to...say we chose the kth row; then we would have ak,1,0677

the first entry in the kth row, times the cofactor of the kth row, first entry,0681

plus ak,2, the kth row, second entry, times the cofactor for the kth row, second entry,0687

up until the kth row, nth entry, and kth row, nth entry cofactor.0694

Similarly, we could have also done this with columns; it would be the first entry, kth column of A,0701

times the cofactor for the first entry, kth column; or the second entry, kth column,0706

with the cofactor of second entry, kth column, up until the nth entry, kth column,0711

nth entry, kth column cofactor.0715

So, that is how it works; don't worry--we will see an example that will make this make a lot more sense.0718

Note that this is true for any value of k, as long as 1 ≤ k ≤ n.0722

So, our k has to be somewhere in these m x n; we can't choose a row that is beyond the dimension,0728

or a column that is beyond the size, of our matrix; that doesn't make sense.0733

But as long as we choose a row that is inside of our matrix, and a column that is inside of our matrix, we can choose any one at all.0737

So, this process can be done with any row or any column, and you will end up getting the exact same result--kind of amazing.0742

We won't see why, but it is pretty cool.0748

This usually means that it is in our interest to choose the row or column that has the most zeroes,0750

because it is really easy: 0 times a cofactor--we don't have to worry about what the cofactor is.0755

It is just immediately going to eliminate itself.0761

So, the thing with the most zeroes, the row or column that has the most zeroes (or the smallest numbers,0763

if we don't have that many zeroes) will help make calculation easier; so that is something to stay on the lookout for.0768

All right, determinant of larger matrices: let's actually put "rubber to the road" and see how this works.0775

First, we notice that there is a 0 here; I like going horizontally, so let's work out this way.0782

Now, notice: a 3 x 3 sign pattern (put it inside of vertical lines, just so we are reminded that we are doing a determinant)0787

is going to look like this: so let's work on this horizontal line here.0802

The first entry in this row is 1; we then go out; we cross out the things on a line with that.0809

That would bring us to the entry 1, times the sign for that cofactor, so -1, times the minor, 2, 3, 3, 5.0819

Next, it is going to be a +; our next one in the row is going to be the 0.0836

We cut out...we don't even really have to care about the cutting out, because 0 times whatever0843

we end up having inside for that minor--that is going to get knocked out, so it doesn't really matter.0849

That is the beauty of choosing the 0.0853

Next, we have the -8; so -8 knocks out what is there; -8, and we are on this one, so - -8, times the minor0855

that is produced by cutting around that -8, cutting a vertical and a horizontal on that -8: 6, 2, -7, 3.0867

We work these out; we have -1 times...down diagonal, 2 times 5, minus up diagonal; 10 - 9 becomes -1.0876

Minus, that becomes + 8 times...6 times 3 becomes 18, minus...-7 times 2 is -14; that cancels out.0889

So, we have -1, negative and negative; that becomes positive 1...0904

Oh, sorry, that did not become negative back here: 2 times 5 is 10, minus 3 times 3 is 9, so we have 10 - 9 is 1; sorry about that.0911

So, that should have been a 1; here is a -1, so this comes out to be -1; it does not cancel out; I'm sorry about that.0921

Then, + 8 times...18 + 14 becomes 32, so -1 +...8 times 32 is 256; so we end up getting 255 as the determinant for this matrix.0928

Alternatively, we could have chosen a different row or a different column.0946

For example, we could have just gone along the top, like this, and we would have had 6 times...0949

and it would be positive, if we are going along the top 6 times...we cross out around it; 0, -8, 3, 5.0955

And then, the next one is minus, -2, times...the minor around that 2 here would cross out to be 1, -8, -7, 5.0964

And then finally, + (because it is a plus in our signs) 3 times 1, 0, -7, 3.0976

You can work it out that way, as well, and you would end up getting 255, as well.0985

I like this row here, because we had that 0; and so, it just managed to knock itself out, right from the beginning.0989

That is that much less calculation for us to have to deal with; I think that is nice--less calculation makes it easier.0995

All right, there is an alternate method for finding the determinant of a 3 x 3 matrix that some people teach.1001

Personally, I want to recommend against using this method--I don't really think there is a good reason to use it.1008

The method we just did, that method with the cofactor expansion, while it seems a little complex at first1014

(it is a lot of things going on) will work for any size matrix at all.1019

And to be honest, this alternate method doesn't actually go any faster, I don't think.1023

So, I would say to try to stick to the cofactor method; I think it works better in general.1029

It gives you the ability to cancel out a whole bunch of zeroes, if you see a bunch of zeroes.1034

And you can use that same method for any size matrix and work down to smaller things.1038

That said, you might have to know it for class, or you might just really want to use it.1043

So, if you must know it, here it is.1046

The first thing you do: you begin by taking the first two columns of the matrix, and you repeat them on the right of the array.1049

So, we have 6, 2, 3, 1, 0, -8, -7, 3, 5; that shows up here, just like normal.1054

But then, we take the first two columns, and then we also repeat this on the right side.1062

So now, we have this extra-large array of numbers.1069

Once we have that array of numbers, we can work with it.1073

We multiply each red down diagonal--we multiply these together, and we add them up.1076

In this case, we would have 6 times 0 times 5; 2 times -8 times -7; and 3 times 1 times 3; that is what we get out of there.1082

And then, we subtract by each of the up diagonals, those blue ones, multiplied together; you subtract by those.1091

Minus (it is always going to be minus), and then -7 times 0 times 3, and then minus1098

(we are subtracting again) 3 times -8 times 6, and then minus 5 times 1 times 2.1105

You work that all out and do a bunch of calculation; you end up getting the exact same number, 255.1112

So, it is an alternate way to find the determinant; it will work if you have a 3 x 3 matrix.1116

It is not that bad; but I don't really think there is a whole lot of reason to use it.1121

It doesn't really go that much faster; you basically have to deal with the same amount of arithmetic.1125

And it is a very specific trick for something that you might have to do on a larger scale, and you can't use that trick anymore.1129

So, I would recommend using the method we were just talking about, with cofactors and minors.1135

But if you really want to use this one, here it is.1139

All right, we are ready to finally see the inverse of a 2 x 2 matrix.1142

So, if we have some 2 x 2 matrix, A = a, b, c, d; then the inverse of A, assuming that the determinant of A is not equal to 0,1145

(if the determinant of A is equal to 0, then we can't invert it at all), then A-1 = 1/ad - bc, times the matrix d, -b, -c, a.1152

So, notice: what we have done there is flipped the location of the diagonal here, and then we put negatives on the b and the c.1168

That is what we are getting here and here and here and here.1176

That is one way of looking at what is going on.1181

Equivalently, you could also write this as 1/detA, because the determinant of A is just ad - bc.1183

So, detA is the exact same thing; and then we are going to end up having the same matrix here and here.1190

That is another way to think about it and remember it; that might be a little bit easier.1195

All right, for the most part, at this level of this course or any similar math class,1199

you are probably not going to need to compute the inverse of a matrix that is any larger than a 2 x 2.1203

You are almost certainly not going to need to do that by hand.1208

But your teacher might want you to; you might just be curious about it.1211

So, if for some reason you need to calculate the inverse of a matrix that is larger than a 2 x 2 matrix,1214

and you have to do it by hand, we will go over a method for this after the examples.1219

We will talk about that after the examples; we will see something for doing that.1223

There is...notice, I said "by hand"; it turns out that if you have a graphing calculator (or access to the Internet),1227

you can actually just plug in matrices and have other computers invert them for you.1233

It is a very useful thing, because the arithmetic of it is very simple, but tedious, and there is a lot of arithmetic.1238

So, we will talk about that a little bit more in the next lesson.1244

Or we will talk about how there are calculators and matrices interacting together.1247

But that is something to think about, if you have to take the inverse of a matrix that is really large;1250

but you can not do it by hand--you are not required to show all of your work by hand--you might want to just use a calculator.1255

That is something to think about.1262

All right, how do you use inverse matrices?1264

If you have some A and A-1, then we know that A-1 times A times B equals B.1266

A-1 and A cancel each other out, and they have no net effect.1272

This is because A-1 times A equals the identity matrix, which is equal to A times A-1.1276

So, if you have the inverse to a matrix, you can multiply on the left side or the right side, and it will create the identity matrix.1281

It creates the identity matrix I, which as we noted in the previous lesson has no effect in multiplication.1287

A-1A up here becomes I; and then I times B--well, the identity matrix times anything becomes just what we already had.1294

So, we get B; so that is why A-1 and A are cancelling out.1301

They turn into the identity matrix, and then that just doesn't do anything.1304

I want you to notice that we can multiply from the left side or the right side.1307

It doesn't matter; it will cancel out in either direction.1311

That is one of the nice things about inverses; they actually will commute, unlike pretty much everything else with matrices.1313

It is important to note that, if we multiply an equation by a matrix on both sides, we have to choose a direction to multiply from1319

and do the same for both parts of the equation.1326

So, if we multiply from the left, we have to multiply from the left on both sides.1329

If we multiply from the right, we have to multiply from the right on both sides.1333

This is because pq is not equal to qp, in general.1336

Multiplying on the left by p is generally very different than multiplying on the right by p.1340

So, if we are going to keep up equality, we have to do the same action; we have to multiply from the left on both sides,1344

because multiplying from different sides is actually a different action with matrices.1349

So, you have to make sure that you multiply from the same side if you want to keep the equality of the equations.1353

So, for example, if we have that A = B, then we can have CA = CB, where we multiply on the left for both sides.1358

Or we could have AC = BC, where we multiply on the right for both sides.1366

But usually, in general, CA is not going to be equal to BC, where we multiply on the left for one, and we multiply on the right for the other.1372

It is in general not going to end up being true; so you will have lost your equality.1380

So, make sure you notice that sort of thing; be careful here--it is dangerous.1384

It is really easy to make this mistake, because so often, when we think about multiplying numbers and equations,1390

like x = 10...we might multiply 3x = 10 times 3, but that is not how it can work in matrices.1395

The only reason we can get away with that in a normal equation is because they commute, so it doesn't matter which side we multiply from.1400

But with matrices, it matters which side we multiply from; so we can't have CA = BC; we have to make sure it is either CA = CB or AC = BC.1408

We have to make sure that we are multiplying both on the left or both on the right.1417

All right, we are ready for some examples.1421

What is the determinant of this 3 x 3 matrix? -2, 1, -3, 4, 2, 0, -1, 0, 1.1423

Our very first thing that we want to do is make a sign marker, just so we can see where all of the signs show up.1429

So, at this point, we need to choose some row or some column to work with.1440

We could choose the top one; that would be fine, but it doesn't have any 0's in it.1446

It has some numbers that are larger than that; so I like this one, because it has -1, 0, and 1.1449

So, one of them is going to cancel out, and the other ones have very little effect on the numbers.1454

Let's work with that: -1 will cancel out those; so we have -1 times 1, -3, 2, 0.1458

Then, the next is still that, because that corresponds to that sign right there...1472

Next, we have minus, because it corresponds to that one, 0, times...and we could figure out what this is,1480

but it doesn't matter; because it is 0, it is going to knock itself out automatically.1490

0 times anything is going to come out as 0, so we don't even have to worry about computing it.1494

And then finally, the 1: that will knock out these, so we have a + here, + 1, times -2, 1, 4, 2.1498

We calculate this; we have -1 times...1 times 0 is 0; 2 times -3 is -6; but it is minus that;1512

so 1 times 0 is 0; minus 2 times -3, so it is a total of +6.1521

And then, plus...1 times...just figure out what this is...-2 times 2 is -4; minus 4 times 1, another -4; so we have a total of -8.1529

We work this out; we have -6 - 8; and we get -14.1540

There are many ways to have done this; we could have also chosen to do this based on this column here.1547

Really quickly, we would have had -3, since we are starting here.1551

We start at positive, but it starts at -3; so -3 times 4, 2, -1, 0,1557

minus...our next sign...0 times...we don't even have to care about it, because it will just knock itself out... plus 1 times -2, 1, 4, 2.1564

Or we could have gone from a different place entirely.1576

We could have also had this, and this would be equal; all of these ways will end up coming out to be the exact same thing.1579

That is one of the cool properties of the determinant.1583

-2 times 2, 0, 0, 1, minus 1 times 4, 0, -1, 1, plus -3 times 4, 2, -1, 0.1587

There are many different ways to do this: this here is the same as this here, is the same as this here.1606

They all end up being equal to -14; so the question of how we want to approach this--1614

which row, which column--we just choose whichever one seems easiest to us.1620

And even if we end up choosing the wrong one--we choose one that is slightly harder--1624

it doesn't matter, because they all come out to be the same thing.1627

We might have to do a few more extra arithmetic steps, but in the end, we will still get the same answer; so it is OK.1629

You don't have to really worry about that.1634

All right, what is this one? We have a 4 x 4, so at this point, we have to take the determinant of this.1636

The first thing we want to do is get a nice sign grid, so we can see all of our plusses and minuses.1641

+ - +...always a positive in the top left...- + - +, + - + -, - + - +; great.1646

So, at this point we want to figure out which is our best row or column to choose.1656

I see two zeroes on this column; so to me, that looks like it is going to make it easiest; I am going to go with that one.1660

I have the 2; it crosses out these; that corresponds to this +1 here, so I just have 2 times...1666

I cross out those other ones; I am left with -1, 3, 0, -4, 5, 4, 1, 1, 0.1676

OK, and then 0 here and the 0 here...we don't even have to worry about them,1684

because they are just going to multiply out to cancel out entirely.1689

So, we only get to having to worry about the 3; that leaves us here.1692

So, it is minus 3 times what gets crossed out: 3 times 2, -1, 3, 0, -4, 5, -3, 1, 1.1695

OK, at this point, let's figure out, of these new ones, which ones we want to use.1714

Let's make a new, smaller, 3 x 3 sign grid, so we can think in terms of that now.1719

OK, so this one...what seems easiest to me is this column...and I would say this row here.1726

We will work with those: we have 2 times whatever the determinant of that larger 3 x 3 is (this one right here);1732

we are working with the 0, so the 0 is going to just knock things out;1741

the only one that we really have to care about is this 4; it will be 4 times...1744

oh, wait, 4 here is there; so we have a -4; we always have to pay attention to that cofactor1750

bringing either a plus or a negative; that is why we make these sign grids here and here.1755

So, we have to pay attention to cofactors.1760

-4 times...that would cross out these things, so...-1, 3, 1, 1.1762

And then, over here, minus 3; so we chose this one, so we are going to have this row starting here: -3 times -0...1773

you don't have to worry about that one; plus...-4 crosses out the other ones1782

that it is horizontal and vertical on...2, 3, -3, 1 is what is left there;1791

And then, minus crosses out, and we get 2, -1, -3, 1.1799

All right, we start working these out; since they are 2 x 2 matrices, we can just work them out now.1812

So, we have 2 times -4 times...-1 times 1 is -1, minus 1 times 3 is -4.1816

Then, minus 3 times -4 times 2 times 1 (is 2), minus -3 times 3; so 2 - -9 gets +11.1826

2 - 3(3)...we have -4 times 11, minus 5 times...2 times 1 is 2, minus 3(-1) 2 here, and then minus 3...1846

-3 times -1 becomes positive 3, but we are subtracting by that, so it is 2 minus 1...2 minus we get -1.1868

OK, so keep working that out: 2 times -4 times -4 is going to come out to be 2 times +16...1882

minus 3...-4 times 11 is -44; these cancel out, and we get + 5; 2 times 16 is 32, minus 3...-44 + 5 is -39.1893

These negatives cancel out; at this point we have this equal to 32 + 3(39) is going to be the same as 3(40) - 3, so + 117.1915

32 + 117 comes out to be 149; so the determinant of our matrix is equal to 149.1929

Great; so by carefully choosing which row we decide to work with, we can make this a whole lot easier.1942

By choosing that third row down, we were able to get a 0 to show here and a 0 to show here,1947

which allowed us to cancel out all of the things, so we only had to figure out two 3 x 3 determinants,1953

which is a lot easier than having to figure out four of them or more--anything like that.1957

So, by carefully choosing the row or column that you do your cofactor expansion on, you can make things a lot easier on yourself.1961

The third example: Prove that, for any 2 x 2 matrix A, where the detA is not equal to 0,1967

then A-1 = 1/(ad - bc) times the matrix d, -b, -c, a.1972

One thing that should be written here is that A is going to be equal to our standard form for just writing a general one, a, b, c, d.1977

So, how would we prove this? Well, we just prove it by showing that A times this supposed A-11985

does, indeed, come out to be the identity matrix, because that is what it means to be the inverse.1991

That is that something times its inverse comes out to be the identity matrix.1995

Some matrix times its inverse matrix comes out to be the identity matrix.1999

That is what it means to be an inverse for matrices.2002

So, let's just check that: let's say A-1 times A.2005

We don't know for sure that it actually is going to turn out to be the inverse, but let's try it.2011

We were told that the detA is not equal to 0; it is the determinant of A...2015

Well, remember: if this is our A right here, then the determinant of A is going to be equal to ad - bc.2019

This would be our only worry in creating this A-1: 1/(ad - bc)--if it is dividing by 0, everything blows up.2026

But since we are told that the determinant of A (which is equal to ad - bc) is not equal to 0,2033

we know that we don't have to worry about dividing by 0, so we can move on.2037

A-1 times A: we have 1/(ad - bc), times the matrix d, -b, -c, a.2040

And then, times A is a, b, c, first, we work through with matrix multiplication.2053

We have our 1/(ad - bc); we will scale later; right now, it will be easier to just work with just the variables, without that fraction getting in the way.2063

So, the first column: we know we will get out to a 2 x 2 matrix in the end; so first row times first column:2074

d times a...actually, let's expand this even more...minus b times c; great.2083

The next one: d times b, minus b times d.2093

The second row on the first column now: -c,a on a,c; -c on a gets us -ca; a on c gets us + ac.2102

The last one: -c,a on b,d gets us -cb + ad.2111

So, we see this; and we do a little bit of simplification, moving things around.2119

Well, db - bd...since b and d are just real numbers, they are commutative, so db - bd just cancel each other out.2124

-ca + ac: once again, they knock each other out.2134

We can rearrange things a little bit; so we have 1/(ad - bc) times the matrix.2137

Well, da - bc is the same thing as ad - bc; this is 0, and this is 0; and -cb + ad...well, we can write that as ad - bc.2142

So, 1/(ad - bc) times this...well, we will get 1, 0, 0, 1, which is exactly what we were looking for.2153

So, this is, indeed, equal to identity matrix; and if we were to do it the other way,2162

A times A-1, to multiply our inverse from the right side, it would end up coming out the same; we would get the same answer.2166

And it turns out that, if you find a matrix that works on one side, you know that it has to work on the other side.2173

But that is a little bit of a deeper result that we haven't talked about explicitly.2178

But you could prove this just by hand, if you wanted to show AA-1; but that is pretty good.2181

The final example: Given that B = -2, 3, 0, 4, and AB = -6, 29, 4, 22, find the matrix A.2186

How are we going to do this? We don't know what A is.2194

We know what AB is; we know what B is; well, notice that we can create a plan like this:2196

AB = AB--that is kind of obvious, but it is true.2201

So, if we came along, we could knock out that B with B-1, so we could have AB = AB,2206

and then we would come along and hit it with B-1 on both sides.2216

And now, we could rewrite this as A =...well, we could cancel out to A on the right side,2221

but we could also see that it is just AB times B-1.2227

We know AB; we know B; and so, if there is a B-1, we can figure out what it is from our B.2235

So, our first step is to figure out what B-1 is.2243

And then, once we know what B-1 is, we just have AB times B-1, and we will have our A.2246

So, that is our theoretical understanding; now it is time to just do the arithmetic.2252

If B = -2, 3, 0, 4, then B-1 equals 1 over the determinant, which is ad - bc,2256

so -2 times 4, minus 0 times 3; so that is -8; times...we flip the location of the main diagonal,2265

and then we put negatives on the other ones: -3 and -0 (we can write as just 0).2274

Simplify that just a little bit to -1/8 times 4, -3, 0, -2.2279

Great; so at this point, we know, from what we showed here, that A is equal to AB times B-1.2287

Well, we know that AB is -6, 29, 4, 22; and B-1 is -1/8 times 4, -3, 0, -2.2299

So, I think it is easier to bring the fraction in afterwards; so let's pull the fraction to the front.2320

The fraction there is just a scalar, so it is just going to scale the matrix.2324

We can scale the matrix any time we want; let's just pull it out to the front, so we can have our matrices do their multiplication.2328

We have -6, 29, 4, 22; 4, -3, 0, -2; OK.2334

There is still that fraction up at the front: -1/8 times whatever comes out of this.2348

It will come out to be a 2 x 2; -6, 29 times 4, 0: -6 times 4 gets us -24; 29 times 0 is just 0.2354

-6, 29 on -3, -2; -6 times -3 gets us positive 18; 29 times -2 gets us -58; so that gets us a positive 40.2368

Oops, I'm sorry; it is not positive; 29 times -2 got us -58, so it is a -40; I'm sorry about that.2382

4, 22 on 4, 0: 4 times 4 gets us 16; and 22 times 0 is just 0.2388

4, 22 on -3, -2: 4 times -3 is -12; 22 times -2 is -44; -12 - 44 comes out to be -56.2396

So, at this point, we can use our -1/8; we simplify this out: we get -1/8 times -24 will become...24/8 is 3; the negatives cancel out, so we get +3.2407

-1/8 times -40 becomes positive 5; -1/8 times 16 becomes -2; -1/8 times -56 becomes positive 7.2419

We have A = 3, 5, -2, 7; and there we are.2429

We had to do a lot of arithmetic to get to this point, so let's double-check and make sure that that is the answer.2436

We know that A times B has to be this right here, because we were given AB, right from the beginning.2441

So, let's take a look: what would A times B be?2448

Well, we know what the A that we just figured out is: that is 3, 5, -2, 7;2450

and the B we started with, that we were given, is -2, 3, 0, 4.2456

So, we work this out; the 3, 5 on -2, 0 is going to get us a -6; 3, 5 on 3, 4 is going to get us 9 + 20, so 29.2463

-2, 7 on -2, 0 is going to get us -2 times -2...+4; and -2, 7 on 3, 4...-2 times 3 gets us -6; 7 times 4 gets us 28; add those together--you get 22.2477

And so, that is exactly the AB that we started with; so it checks out--our answer is good.2491

Great; all right, so that completes an understanding of determinants and inverses.2497

We have a great understanding of how that works right now.2500

We will see you at later--goodbye!2503

However, if you want to check out the stuff for if we want to be able to do inverses for larger than 2 x 2,2505

larger than just that simple formula, let's take a look at it--let's look at that.2514

Finding the inverse of larger matrices: for the method we are about to discuss, we will need some techniques we haven't learned just yet.2519

In the first part of the next lesson, we discuss augmented matrices, row operations, and Gauss-Jordan elimination.2525

You will need to be familiar with these things before what we are about to talk about will totally make sense.2532

So, if you haven't already seen these things, go and check them out first, and then come back and watch this part.2537

It is just the first half of the next lesson--actually, probably more like the first third.2541

The method we are about to go over is applicable for finding the inverse of any m x n matrix.2546

If the matrix has no inverse, if it is singular, this method will end up failing.2550

It is normally easier to first check that there is going to be an inverse, before trying to put all of this work into it.2554

Just check to make sure that there is an inverse by getting the determinant,2559

because getting the determinant will actually take much less time that working through this method.2562

It is a good idea to check the determinant first to make sure that what you are doing will actually manage to work out.2566

All right, let's see how to do this: for an m x n matrix A, you begin by creating an augmented matrix with the identity matrix In.2571

So, if we have some A that is 1, 3, -2, -4, then we leave that part the same, and we drop in an identity matrix.2578

Since this is a 2 x 2, this ends up being a 2 x 2 identity matrix, right here.2586

We have 1, 3, 1, 0; -2, -4, 0, 1; we have it split in this middle, where the left side is A, and the right side is the identity matrix.2591

OK, the next step: you start applying the method of Gauss-Jordan elimination.2601

You use row operations to reduce A, that left side, to the identity matrix.2606

The result of the augmented matrix, once you manage to finally get this to be the identity matrix--2612

what you will have on the right side will be the inverse; you will have A-1 on the right side.2617

So, for example, if we have 1, 3, -2, -4, our first step is that we want to turn this into a 0.2622

We are doing Gauss-Jordan: so we take our row 2, and we add 2 times row 1; so 2 times 1 gets us +2;2629

that cancels to 0; 2 times 3 gets us +6 on -4; that goes to 2; 2 times 1 on 0 gets us 2;2636

2 times 0 on 1 is the same as it was before; so we have our new matrix here.2644

We continue with this method; we had 1, 3, 1, 0; 0, 2, 2, 1 on the previous slide.2648

So at this point, we want to turn this into a 1; so we multiply that entire row by 1/2.2653

This becomes 1; 2 times 1/2 becomes 1; 1 times 1/2 becomes 1/2.2659

At this point, we now want to get rid of this; we want to turn this into a 0 to continue with Gauss-Jordan elimination.2665

So, we subtract: we have a 1 here already, so we subtract 3 of row 2: so -3 times this:2670

1 times -3 on 3 gets us 0; and also, 0 times -3 gets us 1; we don't have any effect there.2678

Minus 3 here gets us -2; and -3 on 1/2 gets us -3/2.2686

So, at this point, we have an identity matrix here; this is just an identity matrix, because it is 1's on the main diagonal and 0's everywhere else.2692

So, what we have over here on the right side is our inverse matrix, A.2700

That is our inverse matrix A; we just bring it down and turn that into a matrix, and we have our answer.2706

Finally, you want to check your work; it is really easy to make a mistake in all of that arithmetic.2712

We were doing the simplest possible of 2 x 2; if you have to do this by hand, you are going to have to be at least doing 3 x 3 or larger.2716

So, it is really easy to end up making a mistake in all of that arithmetic.2722

So, make sure to do notations of what your row operations were, what we saw on the left there,2725

what we talk about in the next lesson when we explain this stuff.2729

And also, at the end, once you get to the very end, check your answer;2732

make sure that A-1 times A is equal to the identity matrix,2735

or A times A-1 is equal to the identity matrix.2738

So, for example, we started with A = this, and we figured out that A-1 should equal this.2741

So, we check our work: we multiply the two matrices together.2745

1, 3 times -2, 1: well, 1 times -2, plus 3 times 1...-2 + 3; that comes out to be 1.2750

1, 3 on -3, 2; that gets us -3, 2 + 3, 2, so that comes out to be 0; they cancel out.2758

Next, -2, -4 on -2, 1; -2 times -2 is 4, minus 4 times 2...4 - 4...that comes out to be 0.2766

And then, -2, -4 on -3, 2...3/2 times 1/ -2 times -3/2 gets us positive 6/2, minus 4/2; so we get 1 out of that.2775

Ultimately, it all checks out; we have figured out that this is, indeed, the inverse; it does end up working out just fine.2788

It is a really good idea to check your work at this point, because it is easy to make a mistake when you are doing that much arithmetic.2795

So, if you have to do this stuff by hand, always check your work at the end, because it is going to be a small amount of time2801

compared to the massive amount of time that you spend doing this.2805

And it would be a real shame if it ended up not being true.2807

All right, I hope that gives you a pretty good sense of how all this inverse and determinant stuff works.2810

And we will see you in the next lesson, when we finally get to see an application of just how powerful matrices are--2815

why we have been interested in them; it is because they allow us to do all sorts of really amazing things2820

that make things much easier than they would be from what we are used to so far--some pretty cool stuff.2825

All right, we will see you at later--goodbye!2830