For more information, please see full course syllabus of Math Analysis

For more information, please see full course syllabus of Math Analysis

### Matrices

- 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 A_{m ×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 (a
_{ij}) to denote the i^{th}row and j^{th}column. - With this idea in mind, we can see a matrix as a series of entries represented by various a
_{ij}. 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:

Since i and j can change, aA = [ a _{ij}]._{ij}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:

That is, every entry of A is multiplied by k. [Note that this is just like multiplying a vector by a scalar.]kA = [ k ·a _{ij}]. - 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:

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.]A + B = [a _{ij}+ b_{ij}]. - 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

where cAB = [c _{ij}],_{ij}= a_{i1}b_{1j}+ a_{i2}b_{2j}+ …+ a_{in}b_{nj}. That is, entry c_{ij}of AB (the entry in its i^{th}row and j^{th}column) is the sum of the products of corresponding entries from A's i^{th}row & B's j^{th}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:**0**_{m×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: I_{n}.) Notice that for any matrix A, IA = A = A I. [ I effectively works the same as multiplying a real number by 1.]

__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*.

### 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
- Introduction
- Definition of a Matrix
- Examples of Matrices
- Talking About Specific Entries
- Using Entries to Talk About Matrices
- Scalar Multiplication
- Matrix Addition
- Matrix Multiplication
- Matrix Multiplication and Order (Size)
- Matrix Multiplication is NOT Commutative
- Special Matrices - Zero Matrix (0)
- Special Matrices - Identity Matrix (I)
- Example 1
- Example 2
- Example 3
- Example 4

- 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
- Matrix Addition 13:08
- 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

### Math Analysis Online

### 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

*Just like a vector allowed us to show more information than a single number, a matrix will allow us to show even more information than a single vector.*0053

*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 things*0067

*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 equations*0116

*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 A _{m 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 important*0376

*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 a _{i,j}, subscript ij.*0543

*And that will denote the i ^{th} row and j^{th} column.*0549

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

*So, a _{i,j} is the i^{th} row, j^{th} column.*0558

*So, that means we could talk about a _{1,1}: that would be first row, first column, so we would get 17.*0563

*We could talk about a _{2,1}; that would be second row, first column; so that would be 0.*0570

*Second row, second column is also 0; we could have a _{2,3}, 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 a _{2,3}.*0585

*Or we could have a _{3,2}, 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 a _{i,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, a _{i,j}.*0647

*So, it is like saying, "Here is some a _{i,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, a _{i,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 i ^{th} row, j^{th} 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 a _{i,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 i ^{th} row, j^{th} column of the resultant matrix will end up being a_{i,j} + b_{i,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 column*0875

*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 see*0917

*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 matrix*0979

*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 i ^{th} row, j^{th} column of AB becomes c_{i,j},*1000

* where c _{i,j} is equal to a_{i,1}b_{1,j} + a_{i,2}b_{2,j},*1007

*up until we get to a _{i,n}b_{n,j}.*1013

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

*a _{i,1} is the i^{th} row of A, first entry.*1018

*The b _{1,j} is the first entry of the j^{th} column, because it is the first row, but in our j^{th} column.*1029

*So, it is the first entry; so it is the first entry, i ^{th} row, A; first entry, j^{th} column, B.*1035

*a _{i,2} is second entry, i^{th} row of A; and b_{2,j} is second entry, j^{th} 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 n ^{th} entry of i^{th} row of A,*1052

*and the n ^{th} entry of the j^{th} column of B.*1057

*Notice that the n ^{th} 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 i ^{th} row has to stop at the n^{th} entry,*1065

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

*The same thing with b _{n,j}: the n^{th} entry in the j^{th} 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 c _{i,j} of AB, the product of the two.*1088

*The entry in its i ^{th} row and its j^{th} column is the sum of the products*1092

*of corresponding entries from A's i ^{th} row and B's j^{th} column.*1099

*So, we are looking at the i ^{th} row of our first one--our first matrix, A, its i^{th} row--*1103

*times the j ^{th} 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 i ^{th} row and j^{th} 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 i ^{th} row (this is the i^{th} row of A),*1135

*and then here--this would end up being the j ^{th} column of B.*1144

*Our first matrix's i ^{th} row, times the second matrix's j^{th} column: we multiply them together,*1154

*where a _{i,1} is multiplied times b_{1,j}, plus a_{i,2} times b_{2,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 c _{i,j}, which is the i^{th} row and j^{th} 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 i ^{th} 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 j ^{th} 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 i ^{th} row, dotted with the j^{th} 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 out*1501

*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

*We have talked about this a little bit so far.*1530

*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 AB _{m 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 question*1599

*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

*So, we have never had to worry about it.*1751

*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

*What about this one here, where we flipped the order of multiplying them?*1854

*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 subtracted*2023

*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 matrix*2050

*(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 I _{2}, 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 I _{5}, 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 be*2134

*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 go*2716

*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 2*2730

*(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 be*3236

*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

1 answer

Last reply by: Professor Selhorst-Jones

Wed May 6, 2015 11:39 AM

Post by enya zh on May 5, 2015

For "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?