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 1 answerLast reply by: Professor Selhorst-JonesWed May 6, 2015 11:39 AMPost by enya zh on May 5, 2015For "Talking About Specific Entries", what if you had to talk about the 21th row and the 51th column? You have to write a2151 but would it be confused with row 215 column 1, or row 2 column 151?

### Matrices

Note: Many teachers and textbooks first introduce matrices as a way to solve systems of linear equations through augmented matrices, row operations, and Gauss-Jordan elimination. If you're looking for that, watch the first part of the lesson Using Matrices to Solve Systems of Linear Equations.
• A matrix (plural: matrices) is a rectangular array where each entry is a number.
• For a matrix with m rows and n columns, we say it has an order of m×n (This property is sometimes called `size' or `dimension'). We can also write order as Am ×n. If a matrix has equal numbers of rows and columns (m=n), we call it a square matrix.
• Matrices are usually denoted by capital letters.
• Two matrices A and B are equal if they have the same order and all their entries are equal.
• We can also talk about some specific entry in row i and column j (where i and j are standing in for numbers). As we use capitals to denote a matrix (A), we often use the corresponding lower case to denote its entries (a). We can talk about a specific entry by using the subscript ij on the letter (aij) to denote the ith row and jth column.
• With this idea in mind, we can see a matrix as a series of entries represented by various aij. This means instead of having to write the entire matrix out (like above) or just using a letter to denote the whole thing (A), we can refer to it by using a single representative entry to stand in for all entries:
 A = [ aij ].
Since i and j can change, aij is a placeholder for all of the entries in A.
• Given some matrix A and a scalar (real number) k, we can multiply the matrix by the number:
 kA = [ k ·aij ].
That is, every entry of A is multiplied by k. [Note that this is just like multiplying a vector by a scalar.]
• Given two matrices A and B that have the same order (m×n, the number of rows and columns), we can add the two matrices together:
 A + B = [aij + bij ].
That is, we add together entries that come from the same "location" in each matrix to create a new matrix. [Note that this is very similar to adding vectors component-wise.]
• If A is an m×n matrix and B is an n ×p matrix, we can multiply them together and create a new matrix AB that is order m×p, and which is defined as
 AB = [cij],
where cij = ai1b1j + ai2b2j + …+ ainbnj. That is, entry cij of AB (the entry in its ith row and jth column) is the sum of the products of corresponding entries from A's ith row & B's jth column. [The idea of matrix multiplication can be very confusing at first. Check out the video to see a lot of visual references to help explain what's going on here.]
• To multiply two matrices together, we have to first be sure that their orders are compatible. The numbers of columns in the first matrix must equal the number of rows in the second matrix.
• Multiplication in the real numbers is commutative, that is, x·y = y ·x: which side you multiply from does not affect the product. ( 5·7 = 7 ·5,   8(−3) = (−3)8 ). However, matrix multiplication is not commutative in general. That is, for most matrices A and B, AB ≠ BA.
• The zero matrix is a matrix that has 0 for all of its entries. A zero matrix can be made with any order. It is denoted by 0. [If you need to show its order: 0m×n.]
• The identity matrix is a square matrix that has 1 for all its entries on the main diagonal and 0 for all other entries. It can be any order, so long as it is square. It is denoted by I. (If you need to show it is order n ×n, you can denote by: In.) Notice that for any matrix A,  IA = A = A I. [ I effectively works the same as multiplying a real number by 1.]

### 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 0:08
• Definition of a Matrix 3:02
• Size or Dimension
• Square Matrix
• Denoted by Capital Letters
• When are Two Matrices Equal?
• Examples of Matrices 6:44
• Rows x Columns
• Talking About Specific Entries 7:48
• We Use Capitals to Denote a Matrix and Lower Case to Denotes Its Entries
• Using Entries to Talk About Matrices 10:08
• Scalar Multiplication 11:26
• Scalar = Real Number
• Example
• Example
• Matrix Multiplication 15:00
• Example
• Matrix Multiplication, cont.
• Matrix Multiplication and Order (Size) 25:26
• Make Sure Their Orders are Compatible
• Matrix Multiplication is NOT Commutative 28:20
• Example
• Special Matrices - Zero Matrix (0) 32:48
• Zero Matrix Has 0 for All of its Entries
• Special Matrices - Identity Matrix (I) 34:14
• Identity Matrix is a Square Matrix That Has 1 for All Its Entries on the Main Diagonal and 0 for All Other Entries
• Example 1 36:16
• Example 2 40:00
• Example 3 44:54
• Example 4 50:08

### Transcription: Matrices

Hi--welcome back to Educator.com.0000

Today, we are going to talk about matrices.0002

In some way, matrices are a natural extension of vectors.0004

Consider that we can express a vector as a horizontal array of numbers, where an array is just a bunch of different spaces to put numbers.0007

So, each component from a vector would be an entry in that array of numbers.0014

So, if we had some vector, (5,47,-8), we could also put that as 5, and then a little bit of space, and then 47, and then a little bit of space, and then -8.0018

We have this array that is three different locations for numbers to go, this rectangular array.0026

A matrix takes this idea and expands on it.0031

The vector was just a single line--it was just a single row going on.0034

Instead of just having columns of numbers (we had that single row with three different columns),0040

we can take that, and we can have rows and columns.0044

This allows us to show lots of information in a single array.0050

So, it is a way to compact lots of information in this single, really useful thing.0060

And we will end up seeing how they are useful later on.0064

Matrices have a huge number of uses, both in math and other fields--they are really, really useful things0067

for science, computer science, engineering, business, economics...so many things.0073

But it is going to take a couple of lessons before we can see how useful they are,0079

because we have to just get the basics of how they work learned before we can really see an application.0082

But in two lessons, we will see how ridiculously easy they make it to solve linear systems.0087

So, once we have matrices learned, and have a good understanding of them, we will be able to solve linear systems easily, which is really cool.0093

Also, I want to mention: this lesson right here is going to be on how matrices work, what a matrix is, and how they interact in various different ways.0101

But many teachers and textbooks don't start with matrices as how a matrix works;0111

they start with it as specifically using it to solve linear equations0116

through augmented matrices, row operations, and Gauss-Jordan elimination.0120

If that is what you are looking for, you have a math class and you are trying to get more understanding of those things,0125

you are going to want to take a look at the lesson Using Matrices to Solve Systems of Linear Equations.0130

The first half of that lesson will go over augmented matrices, row operations, Gauss-Jordan elimination...0135

And there will be some examples about how that stuff works there.0141

So, if that is what you are looking for right now, you might want to go check that out, as opposed to this lesson.0143

However, that said, you are going to end up coming back to everything that is in this lesson.0147

It is just a question of if you introduce the idea of matrices through that stuff first, and then go on to talk about how they work;0151

or if you talk about how matrices work, and then you get to that stuff later.0157

I prefer talking about matrices first, and then getting to the applications; but it depends, from teacher to teacher and textbook to textbook.0160

So, in your case, you might be interested in watching that lesson first.0166

But you are going to end up coming back to this lesson and watching it anyway.0168

And some of the stuff in that lesson will make more sense if you watch this one first.0171

So, you might even find it worthwhile to watch this lesson before you get around to watching that, if you have time.0174

All right, let's get into this: a single object is a matrix, but if we are talking about multiple of them, in the plural, it is matrices.0178

It is a rectangular array where each entry is a number.0189

So, an array is just...imagine a bunch of boxes stacked together to make a rectangle of boxes.0191

And inside of each box, you can put in numbers; so you can put a number here, a number here, a number here, a number here, a number here...0198

And we call each of those places where you could put a number an entry.0204

We have some a = some number here, some number here, going all the way down to some number here,0208

some number here, some number here, going all the way down to some number here;0214

and the same thing going right as well...and then there are a bunch of numbers in the middle.0217

So, it is just a rectangular array--a bunch of places to put numbers in this nice rectangular thing.0222

It is like looking at a piece of grid paper and boxing off some part of it, and then writing a number inside of each of the grids.0228

All right, for a matrix with m rows and n columns (notice that it has m rows and n columns), we say it has an order of m by n.0237

We can put these two things together to talk about the order of the matrix.0255

This property is also sometimes called size or dimension; those are sometimes used as synonyms for order.0259

For the most part, we will just use "order" in this course, but I might say "size" occasionally.0264

We can also write order as Am x n: we write a little part underneath it, m x n.0268

So, if we mainly want to talk about some specific matrix, A for example, we can talk about A.0274

But if we want to mention its order as we are talking about A, we can write its order down to the side as a subscript, m x n.0279

If a matrix has equal numbers of rows and columns, if they are the same number of rows and columns,0286

m = n, we call it a square matrix, because we have a square object.0291

Matrices are usually denoted by capital letters, like A, but you might see other ones, as well.0296

Two matrices, A and B, are equal if they have the same order (they are the same size), and all of their entries are equal.0303

They have the same size, and then, if we go to any given one of the locations over here, it is the same as the same location over here.0310

We go to some location here; it is the same as the location over here.0317

You choose one location here; it is the same thing as this location here.0320

So, they have to look exactly the same for them to be equal to each other.0323

I also really want to drive home this fact that it is an m x n matrix with rows by columns.0327

It is always rows by columns; I found this a little bit confusing at first, but I would recommend:0338

the way to think about this, as a row, is something that goes left-right; a column is something that goes up-down.0344

So, whenever we are talking about stuff in math, we normally talk left-right, then up-down,0353

(x,y) when we are talking about rectangular coordinates.0358

So, when we are on the plane, we talk about the horizontal stuff, and then the vertical stuff,0361

which is why we talk about the rows, which horizontal thing we are talking about,0367

and then the columns, which vertical thing we are talking about.0371

It might get a little bit confusing as you work through it.0374

But just always remember: it is rows then columns; this order of rows, then columns, ends up being very important0376

for a lot of stuff--the way we talk about specific entries.0383

So, it is just really important to remember this "rows by columns."0385

The best mnemonic I can offer you is thinking in terms of the fact that rows are left-right; columns are up-down;0389

and we go left-right, then up-down, so it is rows, then columns, rows by columns.0396

But it is something that you just have to remember.0401

All right, with this idea in mind, that it is rows by columns, let's look at a couple of examples.0404

If we have a 3x3 matrix, then that means we have 3 rows, and we have 3 columns.0413

If we have a 2x3 matrix, then we have 2 rows, and we have 3 columns.0421

If we have a 5x1 matrix, then we have 5 rows, and we have 1 column--the same thing for all of them.0428

Also, I want to point out some of the numbers here.0436

We can have just whole numbers, like 17; but it is also perfectly fine to have decimal numbers, like 4.2.0438

We can have negative numbers, like -19; we can also have irrational numbers, like √2 or π.0445

We can have fractions: -5/7, 1/2; anything that is a real number at all can be one of the entries in a matrix.0452

Any number at all can be something inside of a matrix.0459

Talking about specific entries: we can also talk about some specific entry that is in row i and column j in our matrix.0464

And so, i and j are just standing in for numbers; we will swap them out for numbers later, when we need to.0471

In A that is a 3x3, this matrix right here, the entry in row 2, column 3 is 8.0476

So, we go to row 2: 1, 2; so we are on this one here; and column 3: 1, 2, the third column; so we are on this column here.0486

They end up intersecting right here, and so, we have row 2, column 3, is 8.0498

All right, we can expand on this idea: we use capitals to denote a matrix, like A.0510

So, we can often use corresponding lowercase letters to denote the entries inside of it--0517

so, A to denote the entire matrix, and a if we want to talk about some specific entry inside of it.0522

We can talk about a specific entry by using a subscript, ij, where a subscript--here is our number, and then ij,0528

or any subscript, is just numbers that are down and to the right of the number; that is where we have our subscripts.0535

So, we have ij on it; so we can combine those two, and we have ai,j, subscript ij.0543

And that will denote the ith row and jth column.0549

i came first, so that is talking about the rows; j came second, so that is talking about the columns.0552

So, ai,j is the ith row, jth column.0558

So, that means we could talk about a1,1: that would be first row, first column, so we would get 17.0563

We could talk about a2,1; that would be second row, first column; so that would be 0.0570

Second row, second column is also 0; we could have a2,3, second row, third column; so that would be 8.0576

That is exactly what we figured out at the beginning: row 2, column 3, is a2,3.0585

Or we could have a3,2, third row, second column, which gets us 3.0590

So, this gives us another way to talk about where a number is.0597

We can talk about it in terms of this entry, and a subscript to say which of the entries it is.0601

With this idea in mind, we have another way to talk about a matrix.0607

As opposed to a matrix being the entire matrix, or a matrix just being this capital letter that represents it,0610

we can see it as a series of entries represented by this ai,j.0615

There is a first row first column, first row second column, first row third; then second row first column, second row second column, third row second column, etc.0620

It is just a bunch of entries making up the whole thing.0628

With this idea in mind, instead of having to write the entire matrix like this...we don't have to do the entire matrix.0632

We don't also have to just use a single letter to denote the whole thing, like just A.0642

We can instead refer to it by using a single representative entry to stand in for all entries, ai,j.0647

So, it is like saying, "Here is some ai,j that is talking about all of the different things at once."0653

So, we can see what happens to this one that is representing all of them at once.0659

Notice: since i and j can change, ai,j is a placeholder for all of the entries in A.0664

It is not just one thing; it is all of them at once.0670

In a way, it is representing all of them at once by letting us see how something happens to one of them in there.0674

So, i,j is ith row, jth column; so we have another way to talk about a matrix.0680

All right, at this point we are ready to actually talk about how we can do some basic arithmetic with our matrices.0686

Given some matrix A and a scalar (that is to say, just a real number k), we can multiply the matrix by that number.0692

k times the matrix A becomes k times ai,j, that is, each of the entries of our matrix A gets multiplied by k.0699

So, every entry of A is multiplied by k.0708

Notice that this is just like multiplying a vector by a scalar.0711

If we have some vector, and we multiply it by a scalar, then that scalar multiplies on each of the components of the vector.0714

It is scaling the vector; it is multiplying each of the components.0721

So, if we have a scalar, and we multiply a matrix, that scalar multiplies each of the entries,0723

because a matrix doesn't have components; it has entries, because we have to talk about every row.0728

A vector is just a single row, but a matrix is many, so we talk about multiplying all of the entries.0734

So, other than that distinction between entries or components, it is very much the same thing.0739

A scalar on a vector multiplies each of its numbers.0744

A scalar on a matrix multiplies each of its numbers--it is basically the same thing.0747

So, let's look at a quick example: if we have 3 multiplying on the matrix 1, -4, 10; -19, -7, 20;0752

then we have that 3 multiplies on the first row, first column;0759

and that is going to get 3 times 1, which gets us 3; so the same location is now multiplied by 3.0763

3 times -4 gets us -12; the same location is just multiplied by 3.0769

3 times 10 gets us 30; 3 times -19 is -57; 3 times -17 is -21; 3 times 20 is 60; great.0773

Matrix addition: given two matrices, A and B, that have the same order (they have to have the same order;0783

otherwise it won't work--we will see why that is in just a second), we can add the two matrices together.0789

So, A + B: every ith row, jth column of the resultant matrix will end up being ai,j + bi,j.0794

That is to say, we are adding together entries that come from the same location.0803

If this one was from this place over here, and this one was from this place over here, these two different numbers,0809

we add them together, and that comes out to be that new place in our new matrix that we are creating.0813

Note that this is very similar to adding vectors component-wise; it is very much the same thing as when we added vectors.0818

If you add two vectors, you just take the first components, and you put them together;0824

the second components--you put them together; the third components--you add them together, until you get through the entire vector.0828

If we are doing it with a matrix, it is the same thing, except, instead of components, we now have to do it to each of the entries.0833

So, first row, first column entries: you add them together; first row, second column entries--you add them together, until you get done with that row.0838

Then second row, first column entries--you add them together; second row, second column entries--you add them together;0845

second row, third column...etc., until you have made it through all of the rows and all of the columns.0850

You take a given location; you put the things together from that location; that gives you the value for the same location in the new matrix.0854

Let's look at an example: if we have the matrix 4, 8, -3, 7, and 1, 3, 3, 0, we take first row, first column in both of them,0862

so 4 + 1; and that puts out 5 in the first row, first column of our new matrix.0870

The same thing for first row, second column: 8 and 3 are in them, respectively: 8 + 3 becomes 11; first row, second column0875

is the same location as what it just came from, in our new matrix.0884

The same thing over here: -3 + 3 becomes 0; and finally, 7 + 0 becomes positive 7; great.0887

So, we are keeping the location and adding them together, and that is what we get in our new matrix.0895

Matrix multiplication: now, this one is going to be very different.0900

The previous two made sense; they were a lot like what we were used to doing with vectors.0905

You multiply everything with a scalar; you add based on location with addition.0909

Matrix multiplication--this one is going to twist your brain a little bit.0914

So, it is confusing at first; but the applications in a couple of lessons will hopefully make us see0917

why we end up doing this kind of confusing thing, because there ends up being some purpose to this stuff.0923

But for now, we are not going to really have a very good understanding of why that has to be the case.0929

So, we just want to be careful and follow the rules precisely and pay close attention when you multiply matrices.0933

It is really, really easy to make mistakes with multiplying matrices, especially the first couple of times you are doing it.0938

So, you really have to be very careful and pay attention.0944

So, just follow these rules carefully; it is going to be confusing at first, but don't worry.0946

As we work through a bunch of examples, it will make a lot more sense.0950

The formal definition, the first thing that we are going to see, is probably the most confusing thing of all.0953

But as we see it in action, it will start to make a lot more sense.0957

So, just work through it; you will end up understanding this by the time we get to the examples--no problem.0959

All right, if we have some matrix A, and it is an m x n matrix, and B is an n x p matrix, we can multiply them together.0964

Notice that the m here and the m here match up: there are m rows and n columns in our first matrix,0971

and n rows, p columns in our second matrix; so the number of columns in the first matrix0979

matches up with the number of rows in the second matrix; that is an important idea--it will come up later on.0984

We can multiply them together, and we create a new matrix, AB.0989

That is going to end up being m x p, the things that didn't match up.0994

Or they could match up; but they don't have to match up.0998

And we define AB as: AB, the ith row, jth column of AB becomes ci,j,1000

where ci,j is equal to ai,1b1,j + ai,2b2,j,1007

up until we get to ai,nbn,j.1013

What does that mean? Let's look at that a little bit.1016

ai,1 is the ith row of A, first entry.1018

The b1,j is the first entry of the jth column, because it is the first row, but in our jth column.1029

So, it is the first entry; so it is the first entry, ith row, A; first entry, jth column, B.1035

ai,2 is second entry, ith row of A; and b2,j is second entry, jth column of B.1042

So, we multiply those together; we add them together with the other ones.1050

We keep doing this down the line, where it is the nth entry of ith row of A,1052

and the nth entry of the jth column of B.1057

Notice that the nth entry, in both of those cases, ends up being the last entry of that matrix.1061

If A is an m x n matrix, then for our A right here, i,n, well, the ith row has to stop at the nth entry,1065

because it only has n many columns to work its way through.1073

The same thing with bn,j: the nth entry in the jth column has to stop there,1076

because it has only n many rows to work through, to have things there.1081

So, that ends up stopping; and they stop at the same place, which is useful.1084

All right, so that is the entry ci,j of AB, the product of the two.1088

The entry in its ith row and its jth column is the sum of the products1092

of corresponding entries from A's ith row and B's jth column.1099

So, we are looking at the ith row of our first one--our first matrix, A, its ith row--1103

times the jth column of B, our second matrix.1109

We are multiplying them together, based on first entries, second entries, third entries, fourth entries...1115

We multiply them together, and then we add them all up together.1121

And that ends up giving us the value for the resultant product matrix in its ith row and jth column.1124

I know that it is confusing right now; it will make a lot more sense as soon as we start working on examples.1130

So, we can see this visually as taking the ith row (this is the ith row of A),1135

and then here--this would end up being the jth column of B.1144

Our first matrix's ith row, times the second matrix's jth column: we multiply them together,1154

where ai,1 is multiplied times b1,j, plus ai,2 times b2,j...1162

The first entry here is multiplied times the first entry here; the second entry here, times the second entry here;1167

the third entry here, times the third entry here.1173

We multiply them all together like that; then we sum everything up.1178

And that ends up producing ci,j, which is the ith row and jth column.1182

All right, that is what we end up getting here.1193

All right, we are ready for an example.1195

Let's look at how we would find the entry in the first row and third column of the product from the matrices below.1198

If we are looking for the first row, then that is going to be the first row of our first matrix, so 2, -1.1204

Then, the third column: columns are going to come from the second matrix: so third column...1, 2, 3..the third column here: 5, 0.1213

So, the first entries are 2 times 5; so that is 2 times 5, plus -1 times 0; that gets us 10, so 10 is what goes here in the first row and third column.1223

That is what we end up getting; we end up getting this number 10.1246

We are taking that first row, the third column; we are multiplying together in this strange way.1250

We are adding up, and we are plugging that in for the entry in the matrix that we are creating.1256

Now, notice that this bears some resemblance to dot products.1264

We can think of this ith row as being a vector, because it just has a bunch of pieces to it.1267

It has a bunch of components to it, since it is just one dimension in one way.1272

It is just a vector in one way: 2 and then -1.1276

And then, we have this jth column over here; we can think of this as also being a column vector.1280

We have this vector here and this vector here; we are taking the dot product of them: 2, -1 dotted with 5,0.1285

2 times 5 is 10; -1 times 0 is 0; so we get a total of 10.1292

So, we can think of it as being the ith row, dotted with the jth column.1296

If you think that is confusing--if you never really had a very good understanding of how dot products work in vectors--that is perfectly fine.1301

Don't worry about that; just think of it in terms of multiplying and multiplying like this.1307

But if the dot product stuff made a lot of sense to you in vectors,1311

you can think of it as turning this row into a vector briefly, turning this column into a vector briefly,1314

taking the dot product, and then moving on and doing the same thing with new vectors in a sort-of vector set.1318

It is not exactly like vectors, because we are working inside of a matrix.1324

But it is working very much under that same idea of multiplying based on location of entry, and then adding it all together.1327

All right, let's work this whole thing out: we will use red to talk about everything that this first row is going to.1335

What is the size of this going to come out to?1340

First, let's figure that out, so we can draw in bars for where we are going to multiply.1342

This is a 1, 2, 3...so it is a 3 x 2 matrix, because it has 2 columns.1347

And this has 2 rows and 3 columns, so it is a 2 x 3 matrix; so the 2 and the 2 match up here,1354

so it is going to end up coming out over here; our size is a 3 x 3 matrix.1362

And that also makes sense, because in our first matrix, we have three rows; and in our second matrix, we have three columns.1366

So, each of the things that will come out in our product is a way of putting a row and a column together.1372

Three rows; three columns; they end up stacking into a 3 x 3 product matrix.1377

All right, with that in mind, we know that what is going to have to come out of this is a 3 x 3 matrix.1383

So, I will leave enough room, approximately, to put in a 3 x 3 matrix inside of there.1388

The first one: the first row, first column, will give us the location that is the first row, first column in our product matrix.1394

2 times 2 and -1 times -3, then added together: 2 times 2 is 4; -1 times -3 is positive 3; so 4 + 3 is 7.1402

2, -1 on 1, 3 (first row on second column): 2 times 1 is 2; -1 times 3 is -3; add those together, and you get -1.1414

2, -1 on 5, 0: 2 times 5 is 10; -1 times 0 is 0; so we get 10.1424

So, there is our first row, after we have worked through all three columns.1430

The next one; let's use a new color here: 3, 4 on 2, -3; 3 times 2 gets us 6; 4 times -3 gets us -12; so it comes out to -6.1435

3, 4 on 1, 3; 3 times 1 is 3; 4 times 3 is 12; so that gets us 15 when we add them together.1447

3, 4 on 5, 0; 3 times 5 is 15; 4 times 0 is still 0; so that totals to 15.1454

The last one, the final color: 0, 5 on 2, -3; 0 times 2 is 0; 5 times -3 is -15; 0 times 1 is 0; 5 times 3 is +15;1460

and 0 times 5 is 0; 5 times 0 is 0; 0; and that is our final result.1474

So, we are working through, taking a row in our first matrix, then multiplying it against a column in our second matrix.1481

And we are doing location of entry: first entries, second entries, third entries, fourth entries, as many as we have entries.1491

We multiply the location of entries (first entries together, second entries together, third entries together...)--1497

multiply based on that, and then sum up the whole thing; and that is what gets us what comes out1501

as our product for that row number and that column number.1506

It makes a lot more sense after you just end up working with it, after you end up getting some practice in.1510

As soon as you start working on examples like this yourself, as soon as you do some practice homework, it will make a lot more sense.1515

But we will also get the chance to work on another example a little bit later.1520

All right, matrix multiplication and order: to multiply two matrices together, we have to first be sure that their orders are compatible.1524

The numbers of columns in the first matrix must equal the number of rows in the second matrix.1532

The number of columns in the first matrix must be the same as the number of rows in the second matrix.1538

And then, what comes out of it is this m x p; so ABm x p.1544

So, we have n columns in our first matrix, times n rows in our second matrix.1549

Why is this the case? We can just believe this rule, but let's also get a sense for why it is the case.1558

Well, consider this: if I have, say, a 3 x 2 matrix (let's use red, so we can see how it matches here),1563

and then we have something here, something here, something here, something here, something here, something here;1582

notice that if you look at the length of any row, the length of any row is 2.1589

The length of a row is based on how many columns you have, because each column is an entry.1594

If we look at a row, then it is going to span all of those columns; so it is going to be a question1599

of how many times it has something to go inside of the row.1603

Well, that is going to be a question of how many columns are going through that row.1606

So, the number of entries in a row is going to be based on the number of columns.1610

Similarly, if we have a 2 x 3 matrix, then it is going to be 2 rows, 3 columns.1615

If we grab some column, how many entries are going to be in the column?1624

Well, it is how many rows it goes deep.1628

So, the number of rows is going to tell us how many entries are in a given column.1631

Now, the way matrix multiplication works is: it is this thing, the row, times this thing, the column.1637

It is the row times the column; well, this whole thing has to be first entries against first entries,1644

second entries against second entries, third entries against third entries...1650

So, we have to have the number of entries match up.1653

If we have a different length in the row than the column--they are different lengths, row versus column--we are not going to have them match up.1656

This thing doesn't really make sense; so we are required...the idea of this is for the length here to match up to the length here.1664

And that is why we have this requirement: because the length of a row is based on how many columns it has.1670

The length of a column is based on how many rows it has.1675

So, that is why we have to have these matching here.1678

Otherwise, it won't make sense for the way we have this thing defined, because we will have something longer than the other thing;1682

and what do you multiply by then?--because you don't have the same number of entries; it doesn't make any sense.1687

When you take dot products with vectors, they have to have the same number of components for you to be able to take a dot product.1692

It is sort of the same thing going on here.1696

Matrix multiplication is not commutative; this is absolutely mind-blowing,1699

because it is not something that we have seen anywhere else in math at this point, I am pretty sure.1704

So, at this point, we probably want to know what it means to be commutative,1709

before we try to understand matrices not being commutative.1713

Let's look at that: commutative means that x times y is the same thing as y times x--1716

that this operation from the left is the same thing as the operation from the right.1722

x on the left of y is the same thing as x on the right of y; x times y equals y times x.1726

It doesn't matter which direction that x multiplies from; you get the same thing out of it, at least in the real numbers.1733

5 times 7 is the same thing as 7 times 5; 8 times -3 is the exact same thing as -3 times 8.1738

So, that is something we are pretty used to that makes a lot of sense to us.1746

It doesn't matter which direction you multiply from; it comes out to be the same thing.1749

Well, it is time to start worrying about it: matrices are not commutative, in general.1753

That is, for most matrices A and B, AB is not equal to BA.1758

It is totally different if A multiplies on the left side, or if A multiplies on the right side; you will get totally different things.1764

Now, there are some cases when AB will be equal to BA; it is not an absolute, hard-and-fast rule that AB can never equal BA.1770

It is just like 99% of the time that AB will not be equal to BA.1778

Given two random matrices, chances are that they are not going to end up being the same, depending on the order of multiplication.1782

So, you have to pay attention to who is multiplying from which side.1790

You will have totally different things, depending on changing the order of multiplication, usually.1793

There are some cases where it won't be, but for the most part, they are totally different things.1797

So, you can't rely on having x times y equal to y times x, because all of a sudden, it is not equal to the same thing.1801

You are going to have to pay attention to the order that things are multiplying.1807

Let's look at an example to really make this clear.1811

If we have this first matrix, 4, 2, -3, 1, and 3, 0, -5, 2; then we know we are going to get a 2 x 2 matrix out of this, because they are both 2 x 2.1813

So, 4, 2, 3, -5....4 times 3 gets us 12; 2 times -5 gets us -10; so that comes out to 2.1822

4, 2 on 0, 2: 4 times 0 is 0; 2 times 2 is 4; -3 on 1...let's use a new color here; -3, 1 on 3, -5; -3 times 3 gets us -9; 1 times -5 gets us -5; so -14 total.1829

-3, 1 on 0, 2; -3 times 0 is 0; 1 times 2 is 2; OK, so that is what that first matrix came out to be.1846

We have 3, 0 on 4, -3 now; once again, it is going to come out as a 2 x 2 matrix: 3, 0 on 4, -3:1857

3 times 4 is 12; 0 times -3 is 0; so we have 12.1865

At this point, we already see that we are not the same; on the first one we did, that first multiplication,1869

our first row, first column was 2; in the second one, our first row, first column, was 12.1875

2 versus 12 is totally different; we know that these matrices cannot be the same anymore,1881

because one of their entries is different, and that is enough to say that they are not equal.1886

However, let's get a sense for just how different they are; let's look at the rest of this thing.1890

3, 0...the first row, on the second column now...on 2, 1: 3 times 2 gets us 6; 0 times 1 gets us 0; so 6.1894

-5, 2 on 4, -3; -5 times 4 is -20; 2 times -3 is -6; so -26.1902

-5, 2 on 2, 1: -10 + 2 gets us -8.1911

So, notice: these things are totally and utterly different.1916

2, 4, -14, 2 is completely different than 12, 6, -26, -8.1921

This is a case that really helps us see how different these things are.1929

AB is not equal to BA in a single one of its entries; we get totally different things.1934

So, the order of multiplication, if you are multiplying from the left or you are multiplying from the right--that really, really matters.1940

And that is going to affect how we pay attention to doing matrix algebra in the next two lessons.1945

That is something to think about later on.1949

But for right now, you just have to be aware that AB and BA are totally different.1951

Swapping the order of matrix multiplication means you have to do it again,1955

because you have no idea what is going to come out of it until you actually work through it.1958

All right, finally, we have two special matrices to talk about.1962

First, the zero matrix: the zero matrix is a matrix that has--no big surprise--0 for all of its entries.1966

A zero matrix can be made with any order at all.1973

It is denoted by a 0 as bold; however, if you are writing it by hand, normally you can just tell by writing a zero;1975

and people will know, from context, that that 0 is supposed to be a zero matrix, depending on how the problem is working.1982

But if you really want to denote it, you could probably put some underlines underneath it, or something,1987

to show that it is really important--whatever you want to be able to see that it is definitely a matrix.1991

But for the most part, just writing a 0, if it is next to other matrices...people will know what you are talking about.1997

If you need to show its order, you can write it with a subscript of m x n; that tells us that that zero matrix will have m rows, n columns.2002

So, for example, if we had 3 x 3, then we have 3 rows and 3 columns of nothing but zeroes.2010

If we have 5 x 2, then we have 5 rows and 2 columns of nothing but zeroes.2017

For any matrix A, A - A comes out to be the zero matrix, because each of its entries will be subtracted2023

by it entries again, so each entry will turn into a 0; we get the zero matrix.2029

And also, the zero matrix, times A, equals the zero matrix, which is equal to A times the zero matrix.2033

So, the zero matrix, multiplying on some other matrix, by the left or the right, turns it into the zero matrix.2039

The zero matrix, through multiplication, crushes other matrices into the zero matrix.2045

All right, finally: the identity matrix: the identity matrix is a square matrix2050

(it is always going to be a square) that has 1 for all of its entries on the main diagonal, and 0 for other entries.2056

It can be any order, as long as it is a square.2064

It is denoted with the symbol I; so you just write that out like a normal capital I.2067

If you need to show what its order is (and remember, its order is going to have to be n x n,2073

because it has to have the same number of rows and columns; we can't have different numbers there),2077

we can use just I with a subscript of n, because we don't have to say n x n,2080

because it has to be square, so we just use one number, one letter.2084

So, if we want to talk about I2, then that would be a 2 x 2 matrix with 1's on the diagonal, and 0's everywhere else.2088

If we want to talk about I5, the identity matrix as a 5 x 5, then that is 1's on this main diagonal,2099

from the top left down to the bottom right; and it is going to be 0's everywhere else on the thing.2104

Why is this identity matrix useful? For any matrix A, any matrix at all, as long as they match in orders appropriately,2116

and there is always going to be some identity matrix that will match up appropriately with any given matrix,2123

identity matrix A is equal to A, and A times the identity matrix is equal to A.2128

The identity matrix, multiplied from the left, or the identity matrix, multiplied from the right, comes out to be2134

just whatever matrix we had started with that wasn't the identity matrix.2140

The identity matrix effectively works the same as multiplying a real number by 1.2143

5 times 1 just comes out to be 5; -20 billion times 1 just comes out to be -20 billion.2148

The identity matrix works the same way: I times A just comes out to be A; I times C just comes out to be C.2153

So, whatever matrix we have, we multiply by the identity matrix; it is the multiplicative identity.2161

It just leaves it as it normally was; it leaves its identity alone--it leaves it the same.2166

All right, we are ready for some examples.2172

First, a little bit of scalar multiplication: let's do the scalar multiplication, and then we will do the subtraction or addition.2174

2 times 5, -7, 2, 11, 3, 4; its order stays the same, so 2 times 5 is 10; 2 times -7 is -14;2180

2 times 2 is 4; 2 times 11 is 22; 2 times 3 is 6; and 2 times -4 is -8.2189

So, at this point, I am going to change this into a plus, and I am going to say that we had -3 here.2196

+ -3 times something is the same thing as -3 times something.2202

We can pull that negative out and put it on the scalar instead.2205

We do that here: -3 times 3 gets us -9; -3 times -2 gets us +6; -3 times 2 gets us -6; -3 times 6 gets us -18; -3 times 0 gets us 0; -3 times -5 gets us +15.2209

At this point, we are ready to combine them: we combine the two things together.2224

We do it based on location: so 10 and -9 will go in the first row, first column, because they came from the first row and first column.2229

10 and -9 gets us 1; it is going to have the same order here.2235

-14 and 6 gets us -8; -6 and 4 gets us -2; 22 and -18 gets us +4; 6 and 0 gets us +6; -8 and 15; and we have 7; and there is our matrix.2239

All right, now we could have done this a different way.2262

At this point up here, we chose to do plus onto a negative scale, but we could have left it with subtraction.2265

If we had chosen to leave it as subtraction, our first matrix would have remained the same: 10, -14...2272

still the same scalar, so nothing is going to change here from that first matrix.2279

And now, it is going to be minus...we could multiply that scalar by it instead.2283

So, we are going to leave it as a subtraction, but we are just going to multiply that +3 as if it wasn't changed over.2288

So, 3 times 3 gets us 9; 3 times -2 gets us -6; 3 times 2 gets us +5; 3 times 6 gets us +18; 0; and -15.2297

All right, notice: the only difference between these two matrices is this negative sign having hit everything.2308

At this point, we can subtract, and we would end up having 10 - 9; 10 - 9 comes out to be 1.2315

-14 - -6; well, - -6 becomes + 6; -14 + 6 becomes -8; 4 - 6 is -2; 22 - 18 is 4; 6 - 0 is 6; -8 - -15 becomes + 15; -8 + 15 becomes +7.2323

So, we end up getting the exact same thing.2343

Whichever way we do it ends up coming out to be the same thing, which is what we had hoped.2345

I would, for the most part, recommend doing this method that I did here, where you make it addition, and you put the negative on the scalar.2350

You swap it from being subtraction to addition, and then you put the negative on the scalar.2359

And then, you multiply that through, because it gives you one less thing to have to keep track of,2364

as opposed to having to remember the entire time, "I am subtracting; I am subtracting; I am subtracting,"2367

because then, if you forget to subtract just once, your answer is gone; you now have the wrong answer.2371

But if you put the negative on it there, then you remember to multiply by the negative the whole time through.2375

And then, from there, it is just addition.2380

I think it is easier that way; but if you think it would be easier by doing subtraction, go ahead and do that.2382

Whatever works best for you is what you want to use.2386

But I personally would recommend multiplying by the negative, and then doing addition, as opposed to keeping around subtraction.2388

But they will both work just fine.2394

The next example: A is this matrix; B is this matrix; C is this matrix; if the matrix multiplication below is possible,2396

give the order, the size, of the matrix that it would result in.2402

So, we have AB times B...OK, to do that, the first thing we are going to have to do is talk about what each one of these sizes are.2406

If we have 3 rows, 2 columns, that is a 3 x 2 matrix for A.2414

B is 2 columns, 3 rows, so that is a 2 x 3 matrix for B.2420

And C is 3 rows, 3 columns; it is square, so we have a 3 x 2 matrix here.2426

Great; so at this point, it is a question of comparing--do these things match up?2433

AB is going to be 3 x 2, multiplying against a matrix that is 2 x 3.2436

To do this, we have to have...the first one's number of columns has to match the second one's number of rows.2450

But an easier way to do this is to just think in terms of the inner numbers.2455

Are the inner numbers the same? Well, the inner numbers are both 2; so now, what is going to result is the outer numbers.2458

We get those outer numbers as the resultant size of the matrix; so we will get a 3 x 3 matrix in the end.2464

If we reversed this and looked at B times A, then we would have a 2 x 3 matrix times a 3 x 2 matrix.2470

We check: are the inner numbers the same? 3 and 3 are the same, so it becomes the outer numbers; those will be our resultant.2477

So, we will get a 2 x 2; so notice, AB and BA are very different in the end.2483

And we can see that, just based on the fact that they have totally different orders.2487

So, you can end up getting different sizes, as well, based on it.2490

Not only are they not commutative (we can't rely on AB being the same thing as BA); we can't even rely on the size remaining the same.2493

Next, let's look at C on B: that is a 3 x 3 times a 2 x 3; so in this case, do they match up?2503

Does 2 match up with 3--are they the same number?2515

No, they don't match up; so we have no solution here.2517

A on C: a 3 x 2 matrix multiplied with a (sorry, I need to switch to green) 3 x 3 matrix--do they match up?2520

3 and 2 don't match up; so we don't have anything that comes out of that, as a result.2534

And finally, CAB: well, can we multiply multiple matrices together?2539

Sure--we do one matrix multiplication; that comes out as another matrix; and then just multiply the resultant thing.2543

So, let's see if we can work through this: CAB is a 3 x 3, multiplied by a 3 x 2, multiplied by a 2 x 3.2548

So, our first question that we want to do...let's work from the left to the right.2560

So, we will look at what CA became, and then we will multiply by B.2563

So, CA...we have a 3 here; we have a 3 here; so that is going to result in a 3 x 2 (that is what CA would come out as),2567

times B still (we have to do B), 2 x 3; so now we ask--do they match on the inside?2576

They match on the inside, so what is going to result is a 3 x 3; there is our answer.2582

Let's also look at it if we had gone from another direction--if instead of going from the left, we came from the right.2588

We would hope that that would work out, because if it didn't, then there are some issues with how we have this stuff set up.2591

So, let's look at CAB, if we had done CAB from the right side to the left.2596

We have the same thing: a 3 x 3 is C; A is 3 x 2; and B is a 2 x 3.2605

So now, we are working from the right side; so what does AB come out to be?2613

Well, that is a 2 here and a 2 here, so that comes out to be 3.2617

And look, we already did this--we already figured out what AB is.2619

We know that that should come out to be a 3 x 3; so from there, we have a 3 x 3 matrix.2622

And then, what came out of a AB is a 3 x 3 matrix; the 3's match up, so what we get in the end is a 3 x 3 matrix.2628

So, that checks out; either way we did it, it is the same.2638

One last thing to point out here: look, if we had a 3 x 3, if we had CAB one more time...a 3 x 2, and then a 2 x 3...2641

well, what we can do is say, "Do the inner parts match?"2654

The inner part here matches, and the inner parts here match.2658

Ultimately, what is going to happen is that all of the inner parts are required to match for multiplication to happen.2661

But they all disappear; the only thing that ends up making it out in the end is things on the far edges, the 3 and 3 on both sides.2665

So, what is going to come out in the end is a 3 x 3.2674

So, if you have multiple matrices multiplying against each other, you can just check and make sure that all of the inner numbers match up against each other.2677

And then, the size of the resulting thing will end up just being the far edge numbers, which were, in this case, 3 and 3.2683

All right, the next example is a big, big one of matrix multiplication.2691

We are going to work to simplify this, so first, let's see what size our product is going to come out to be.2695

So, we have a 3 x 3 and a 3 x 3; it is possible--no surprise there, since it was given to us as a problem.2700

That is going to come out as a 3 x 3 matrix.2705

At this point, let's work it out; since we are working with a 3 x 3 matrix, we will leave a nice, big chunk of space for us to work inside of.2709

So, we are going to work this out: the first row times the first column will get the number that is going to go2716

in our first row, first column of our resultant product matrix.2726

So, 6 (the first entry of the first row), times the first entry of the first column, -2, added to 22730

(the second entry of the first row), times the second entry of our first column, plus 3 times the final entry,2739

the third entry, of our first row, times the final of entry, the third entry, of our first column:2749

we work that out; we get -12 + 8 - 3; we get -15 + 8, so we get -7.2754

We have -7 for the entry--that first row, first column entry.2765

That is what is going on behind the scenes.2769

We are taking that row; we are taking that column; we are multiplying them based on how the entries multiply together,2772

matching the entries, multiplying matching entries, and then summing up the whole thing.2778

We can see that as I just wrote it out there.2781

So, clearly, this takes a lot of arithmetic: we are doing three multiplications and three additions--it is tough to do this.2785

I would recommend, if you aren't excellent at doing mental math: really try to keep some scratch paper that you are doing on the side.2791

Be very, very careful working with your calculator; it is so easy to make mistakes in matrix multiplication,2798

Especially the first couple of times you are doing it--it is something you really have to be careful with the first couple of times.2803

And it is something you always have to be careful with, because you can always easily make mistakes.2807

Even I will very easily make mistakes with matrix multiplication.2811

But if you just stay focused and pay attention to these rules carefully, and you work carefully,2814

you can always make sure that you get the answer right.2819

But just really be careful with matrix multiplication.2821

It is an easy place to make simple mistakes where you understand what is going on,2823

but you just made a little arithmetic error, and it makes your answer wrong; so be careful.2827

All right, I am going to do the rest of these by just working through them in my head and talking them out,2831

because I am pretty good at this; but be careful when you are doing it.2835

If you are not really good at doing this sort of stuff in your head, be careful; do it on scratch paper.2839

And the larger the matrices get, the harder it is to do in your head.2843

So, 6, 2, 3 on 1, 2, 0; the first row on the second column: 6 times 1 is 6; 2 times 2 is 4; 6 + 4 is 10.2845

3 times 0 is 0; so we get 10.2854

And then the final column: 6, 2, 3 on -3, 0, 1: 6 times -3 is -18; 2 times 0 is 0, so still -18; 3 times 1 is 3; so -18 + 3 is -15.2857

The next one (let's switch to a new color): second row: 1, 0, -8; multiply that by the first row;2868

so, second row, first column, is going to be 1, 0, -8 on -2, 4, -1; 1 times -2 is -2; 0 times 4 is 0;2874

so, -2 + -8 times -2 becomes +8, so -2 + +8 becomes +6.2881

The next one: 1, 0, -8 on 1, 2, 0; 1 times 1 is 1; 0 times 2 is 0; -8 times 0 is 0; so we just get 1 out of that.2890

1, 0, -8 on -3, 0, 1; 1 times -3 is -3; 0 on 0 is 0; -8 times 1 is -8; so -3 + -8 becomes -11.2901

And then finally, -7, 3, 5 on -2, 4, -1: -7 times -2 becomes +14; 3 times 4 becomes +12; 14 + 12 becomes +26;2911

5 times -1 is -5; 26 - 5 is 21; OK; -7, 3, 5 on 1, 2, 0: -7 times 1 is -7; 3 times 2 is 6; -7 + 6 is -1; and then + 5 times 0,2927

so we just come out to be -1 here.2942

And then the final one: -7, 3, 5, on -3, 0, 1: -7 times -3 becomes +21; 3 times 0 is 0; +21 still; plus 5 times 1 is 5; 21 + 5 is 26.2944

OK, so hopefully, that points out just how much arithmetic you are having to do in your head here.2960

I really want you to be careful, because this is the easiest way to make mistakes.2967

And it is the pretty silly way to end up losing points on a test or homework,2970

because it is not because you don't understand what is going on.2973

It is just because you are trying to do so much arithmetic in your head; it is easy to make a mistake.2975

If you end up having any difficulty with something particularly hard, just write it out on paper.2979

Or if you have a really nice graphing calculator, where you can see each of the numbers you are putting into it,2983

be careful; watch; make sure that what you have there matches up to what you have on the paper.2987

And then, make sure that you are being careful if you are using a calculator.2991

So, just be careful, however you are approaching it.2994

The actual process isn't that difficult, once you get used to it; but it is always going to be a real chance of making mistakes,2996

just because there is so much arithmetic going on.3002

All right, the final example: Prove that for any 2 x 2 matrix A, the zero matrix times A equals 0, and the identity matrix times A equals A.3004

So, any 2 x 2 matrix A--it says "any," so that means we can't just use some 2 x 2 matrix.3012

We can't actually put down numbers, because we have to be able to have this true for any 2 x 2 matrix.3018

So, if we came up with some matrix, like 3, 1, 7, 47; well, maybe it happens to be true for that matrix.3023

So, we have to figure out a way to be able to show that it is true for every matrix.3031

We need to write about this in a general form, so we do the same thing that we have everything set up in, where we use variables,3034

because if we have it as a, b, c, d, then any 2 x 2 matrix...we can just swap out a, b, c, d for actual numerical values,3039

and we will have any 2 x 2 matrix--this will be true for any 2 x 2 matrix.3049

So, we have that as an idea; so now we can just try multiplying.3052

If this is going to work for this A here, a, b, c, d, that is a stand-in for any 2 x 2 matrix at all.3056

So, if we can show that the zero matrix times this comes out to be 0 anyway, then it has to be true for all of them,3063

because they are just going to end up swapping out the variables, a, b, c, d, for actual numbers.3068

So, let's try this: the zero matrix times A = 0: well, if we have the zero matrix, then we are going to have 0, 0, 0, 0,3072

because it is multiplying against a 2 x 2; our A is a, b, c, d.3079

This is actually pretty easy matrix multiplication, thankfully.3084

0 and 0 times a and c; look, that is going to crush it down to 0.3087

We know that we are going to have to get a 2 x 2 matrix, because we started with 2 x 2 times 2 x 2.3091

0, 0 on b, d: well, 0 times b and 0 times d--that comes out to be 0.3095

0, 0 on a, c: 0 times a and 0 times c--that comes out to be 0.3100

0, 0 on b, d: 0 times b and 0 times d--no surprise there--comes out to be 0.3103

So, we see, because it has nothing but zeroes that we are multiplying by:3108

whatever it is going to hit is going to get turned into a 0.3111

So, that is why the zero matrix, multiplied by any matrix at all, ends up coming out to be the zero matrix.3113

So, this checks out: and we can see that, if we had put it as the zero matrix,3119

multiplying from the right as opposed to the left, basically the same thing is going to happen,3123

because we are multiplying against, it is going to hit nothing but zeroes, so it is just going to get turned to 0's automatically.3126

So, the zero matrix, multiplied against any matrix, becomes the zero matrix.3132

Next, the identity matrix times A: let's show that that becomes A.3136

The identity matrix: well, A is a 2 x 2, so that means that our identity matrix will have to also be a 2 x 2.3140

So, 1's are on the main diagonal, with 0's everywhere else.3145

1, 0, 0, 1; the main diagonal has 1's, and then everything else will have 0's (in this case, not many 0's).3149

a, b, c, d: this will take a little bit more thought.3156

1, 0 on a, c: 1 times a...we are going to have to get a 2 x 2, because we started with 2 2 x 2 matrices...3160

1 times a comes out to be a; 0 times c becomes 0; so we have a.3170

1, 0 on b, d: well, 1 times b comes out to be b; 0 times d comes out to be 0, so we get just b.3175

0, 1 on a, c: 0 times a comes out to be 0; 1 times c comes out to be c.3183

0, 1 on b, d: 0 times b comes out to be 0; 1 times d comes out to be d.3189

That checks out; it ended up being the same thing.3194

So, it is a little bit harder to see why the identity matrix is working;3196

but basically, what it is doing is: when you multiply some other matrix by it,3199

since it has just a 1 in one place, it is seeing what is at the same location over here.3204

a is at the same location; b is at the same location; so they end up popping out.3209

For this one down here, 0 and 1, it is seeing what is at the same location: c and d are there, so they get to pop out, as well.3213

A similar thing ends up happening if you multiply the identity matrix from the right, instead of from the left.3219

Try it out for yourself--take a look, if you are curious about seeing it.3223

All right, that shows everything that we have.3226

We have a pretty good understanding of how matrices work at this point.3228

We are ready to go and see some of the cool things that we can start doing with them.3230

So, we will talk about some new ideas in the next section, the next lesson.3233

And then two lessons from now, we will see just how powerful these things can be3236

for solving problems that would seem really difficult; we are going to turn them so easy so quickly.3240

All right, we will see you at Educator.com later--goodbye!3245