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Let's do some more matrix multiplication examples,

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because I think it is all about seeing as many examples

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as possible.

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So let's do what may seem to be a more difficult problem,

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and it might not be even clear that we can

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multiply these matrices.

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And maybe that's the first thing we should think about.

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So let's say I wanted to multiply the matrix-- I'll do

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it relatively small so we don't run out of space.

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3, 1, 2, minus 2, 0, 5.

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Let's say I want to multiply that times the matrix, minus

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1, 0, 2, 3, 5, 5.

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I'm just making up these numbers.

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So the first thing you might be wondering is, well can I

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even multiply these matrices?

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Because you know from the first video we did on

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matrices, that you can't add these two matrices.

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This term corresponds to this one; this one corresponds to

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this one; but this term corresponds to nothing over

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here so you couldn't add or subtract these matrices.

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So the question is, can I multiply these matrices?

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Well, what did we learn about multiplying matrices?

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We know that, for example, if this is going to result in

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some matrix--

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However we don't know even what the dimensions are yet

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until we work through this example, although there is a

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quick way for figuring it out.

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So this first term here, the upper left term, where does it

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get its row information from and where does it get its

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column information from?

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Well, it gets its row information from here.

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So it is essentially this row times which column?

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Times this column, right?

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And we can actually take the dot product of this row vector

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and this column vector because they have the same length.

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This is a column vector but it has a length of 3, right?

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But it's a 3 by 1, it has three elements in it.

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And this is a 1 by 3 row vector, but it also has three

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elements in it.

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So we actually can take the dot product or we can

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multiply these two.

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And similarly we can multiply this times this whole thing to

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get this term right here.

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And we can multiply this thing times this thing to get this

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term, and then this thing times that

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term to get that term.

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So it actually turns out that you can-- so what kind of a

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matrix is this?

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Let's call it that this is matrix-- let me switch to

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[UNINTELLIGIBLE].

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So this is matrix A.

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And what are it's dimensions?

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It has 2 rows, 1, 2, and 3 columns.

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So it's a 2 by 3 matrix.

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We're multiplying it times B.

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And what are B's dimensions?

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Well, it has 3 rows, 1, 2, 3.

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So it is a 3 by-- and how many columns does it have?

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1, 2-- 2 matrix.

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So it turns out that we can multiply two matrices.

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You can say that the number-- if, on the first matrix-- the

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number of columns is equal to the number of rows in the

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second matrix.

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So here, 2 by 3 times 3 by 2, we can multiply.

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For example, we could have multiplied, if

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this is matrix C.

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I don't know if I take so much time to keep

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bolding these things.

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And I don't care how many rows it has.

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It can have n rows, n times a columns.

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I can multiply it times matrix D, as long as

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matrix D has a rows.

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As long as you can say these two inner numbers are the

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same, right?

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This 3 is the same as 3.

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And why does that matter, what was the logic?

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Because this row will have 3 elements because there's 3

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columns, and each column vector here will have 3

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elements, because there's 3 rows.

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That's where intuition comes from, but if you had to do it

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really quickly you say 2 by 3, 3 by 2, this number is equal

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to that number, I can multiply.

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So let me clear up some space and let's do the

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multiplication.

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Let's do some multiplication.

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I'm debating where I should do it, actually I think I should

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do it down here maybe because I'll have more space.

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So let me do it down there; I don't have to

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erase anything else.

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So let me get some space ready.

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OK, this will take up a lot of space.

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So to get this row 1, column 1 element, what do I do?

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I multiply this vector times this vector.

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I take the dot product, right?

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So it's 3 times negative 1-- I'm just going to write it all

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-- 3 times minus 1, plus 1 times 0, plus 2 times 2.

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There, we got the first term.

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So the second term here, what am I going to do?

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I'm going to multiply that vector times that row vector

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times this column vector.

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And I think you're getting the hang of this, and really the

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hardest part about this is staying focused and not making

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a careless mistake.

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And not getting it confused with rows and

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columns and all that.

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It just sends blood to your brain, but it's not

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that hard, I think.

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So what do we do?

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We multiply this row vector times this column vector to

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get row 1, column 2, right?

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Because this row, row 1, column 2.

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3 times 3, plus 1 times 5, plus 2 times 5, right.

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We're just multiplying the corresponding terms, the third

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term times the third term, the second term times the second

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term, the first term times the first term.

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Although, in this case they're going down, in this case

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they're going left and right.

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Oh, we add them all up.

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OK, so now we're in the second row, and we get our row

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information from the first vector-- and let me do a red

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that I never use because I think it's kind of tacky, this

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red right here.

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So I'm going to multiply this row vector

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times this column vector.

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So it's minus 2 times minus 1, plus 0 times 0,

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plus 5 times 2.

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We're almost done.

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Let me see-- I don't like this color at all-- and now we're

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going to multiply this row, because we're in this bottom

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row, we're in row 2, column 2, to row 2, column 2.

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So it's minus 2 times 3, plus 0 times 5, plus 5 times 5.

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And then if we simplify, let's see, this is minus 3 plus 0,

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plus 4, so this-- if I have my math correct--

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simplifies to 1.

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9 plus 5 is 14, plus 10 is 24.

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This is 1, 24, and then minus 2 times minus 1 is 2, plus 10,

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so this is 12.

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And then minus 2 times 3 is minus 6, plus 10-- this is 0--

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so minus 6 plus 10 is 4.

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So that's interesting.

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When I multiplied a 2 by 3 vector times a 3 by 2 vector,

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what did I get?

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I got a 2 by 2 matrix.

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A 2 by 3 matrix, a 2 by 3 times 3 by 2 matrix, I got a 2

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by 2 matrix.

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And where do you see a 2 by 2?

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Well, it's like this got multiplied with this, and what

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we have left over is a 2 by 2 matrix.

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So in general-- well actually, before I go into the general,

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let me ask you a question.

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Could I have multiplied the matrices the other way?

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Could I have multiplied-- so this right here that is A

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times B, or you can sometimes write this AB, and we'd bold

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it all up so we know it's matrices.

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So could we have multiplied B times A?

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Let me clear this down here and let's try.

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Let's see if we can multiply B times A.

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I think you can already suspect that, since I'm asking

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the question, maybe you cannot.

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Let's clear up some space.

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Let's try to do it the other way around, let's try to

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multiply B times A.

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So B is minus 1, 0, 2, 3, 5, 5.

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And A is-- I'm switching the order-- 3, 1,

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2, minus 2, 0, 5.

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And I tend to put brackets around my matrices.

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Some people have these big parentheses.

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It's just all notation; there's nothing particular

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about notation.

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So let's see if you can multiply these.

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So we learned that you get the row information from the first

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matrix and the column information

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from the second one.

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So this term, in theory, should be that row times what?

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Well, actually, it turns out that you can multiply them.

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Why?

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Because this is a 3 by 2, and this is a 2 by 3, right?

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So we're going to take that row times what?-- times this

186
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column to get the first term, right?

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So what is it going to be?

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It's going to be minus 1.

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So I actually thought I was doing a counter example, but

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actually because this too is the same as this, or when you

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00:09:22,929 --> 00:09:27,399
switch the row this is the same as this, you

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can multiply them.

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So I wanted to do a counter example, but hey.

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Let's just work through this because it never hurts to see

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another example.

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And you can see that I just do this on the fly.

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So let's do this.

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00:09:34,909 --> 00:09:38,379
And actually ahead of time, how large will this matrix be?

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00:09:38,379 --> 00:09:39,559
Well, this is interesting.

200
00:09:39,559 --> 00:09:42,099
It's actually going to be a 3 by 3 matrix,

201
00:09:42,100 --> 00:09:43,029
a much bigger matrix.

202
00:09:43,029 --> 00:09:45,370
Let's work it all out, and maybe you want to pause it and

203
00:09:45,370 --> 00:09:46,029
try it yourself.

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00:09:46,029 --> 00:09:57,639
This row times this column, so minus 1 times 3, it's minus 3,

205
00:09:57,639 --> 00:10:01,929
3 times minus 2 is minus 6.

206
00:10:01,929 --> 00:10:06,329
And then it's going to be this row times this column.

207
00:10:06,330 --> 00:10:09,980
So it's minus 1 times 1 plus 3 times 0, so that's

208
00:10:09,980 --> 00:10:11,639
just minus 1, right?

209
00:10:11,639 --> 00:10:13,250
Because 3 times 0 is 0.

210
00:10:13,250 --> 00:10:17,100
And then, that was that one, then there's the middle one,

211
00:10:17,100 --> 00:10:19,580
and now we get the row 1, column 3.

212
00:10:19,580 --> 00:10:20,750
So row 1, column 3.

213
00:10:20,750 --> 00:10:23,750
So it's that row times this column.

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00:10:23,750 --> 00:10:27,669
You can tell, this is often better done by a computer.

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00:10:27,669 --> 00:10:32,139
Minus 1 times 2 is minus 2 plus 15-- 3 times 5-- so minus

216
00:10:32,139 --> 00:10:34,939
2 plus 5 is 13.

217
00:10:34,940 --> 00:10:36,330
Let's keep going.

218
00:10:36,330 --> 00:10:39,030
So now we're going to take--

219
00:10:39,029 --> 00:10:42,759
I'm sweating, this is so computationally intensive--

220
00:10:42,759 --> 00:10:45,669
We're taking this row times each of these columns.

221
00:10:45,669 --> 00:10:46,769
And actually we are going to learn later that there are

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00:10:46,769 --> 00:10:49,879
multiple ways of actually thinking about how this

223
00:10:49,879 --> 00:10:52,129
multiplication happens, even multiple ways by computer, but

224
00:10:52,129 --> 00:10:53,840
this is the traditional way.

225
00:10:53,840 --> 00:10:56,070
So this row times each of these columns, right?

226
00:10:56,070 --> 00:11:01,830
So 0, 5, so 0 times 3 plus 5 times minus 2, that's minus

227
00:11:01,830 --> 00:11:06,830
10, and it's 0 times 1 plus 5 times 0.

228
00:11:06,830 --> 00:11:08,840
That's easy, that's 0.

229
00:11:08,840 --> 00:11:14,149
0 times 2 plus 5 times 5 is 25-- almost

230
00:11:14,149 --> 00:11:15,689
there, almost done.

231
00:11:15,690 --> 00:11:18,580
Now we're going to take this row and multiply it times each

232
00:11:18,580 --> 00:11:19,320
of these columns.

233
00:11:19,320 --> 00:11:25,620
So 2 times 3, that's 6, plus minus 10, so

234
00:11:25,620 --> 00:11:27,700
that's minus 4, right?

235
00:11:27,700 --> 00:11:30,330
2 times 3 plus 5 times minus 2.

236
00:11:30,330 --> 00:11:33,470
Yes, that's minus 4, 6 minus 10, right?

237
00:11:33,470 --> 00:11:36,670
We have 2 times 1 plus 5 times 0, that's 2.

238
00:11:36,669 --> 00:11:43,589
Then you have 2 times 2 plus 5 times 5, so 4 plus 25 is 29.

239
00:11:43,590 --> 00:11:48,180
And of course that first term, minus 3 minus 6, so

240
00:11:48,179 --> 00:11:50,639
this is minus 9.

241
00:11:50,639 --> 00:11:51,840
So there you have it.

242
00:11:51,840 --> 00:11:55,160
We multiplied the 3 by 2 matrix times a 2 by 3 matrix,

243
00:11:55,159 --> 00:11:56,959
and we got a 3 by 3 matrix.

244
00:11:56,960 --> 00:11:58,240
And where did that 3 by 3 come from?

245
00:11:58,240 --> 00:12:02,970
Because this 3 is the number of rows in the first matrix,

246
00:12:02,970 --> 00:12:05,720
and this 3 is the number of columns in the second matrix,

247
00:12:05,720 --> 00:12:08,129
which makes sense because we got our row information from

248
00:12:08,129 --> 00:12:10,830
the first matrix and our column information from the

249
00:12:10,830 --> 00:12:12,300
second matrix.

250
00:12:12,299 --> 00:12:14,609
Now let me actually show you an example

251
00:12:14,610 --> 00:12:18,360
that you cannot multiply.

252
00:12:18,360 --> 00:12:22,120
So what if I wanted to multiply a-- let me do a very

253
00:12:22,120 --> 00:12:26,679
simple example-- what if I wanted to multiply

254
00:12:26,679 --> 00:12:30,849
the matrix, 2, 1.

255
00:12:30,850 --> 00:12:33,430
And really, all this is, is a row vector.

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And let's say that I wanted to multiply this times-- I don't

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know, so this is a 2 by 1.

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So then let me say I want to multiply this times-- so let

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me think of something-- 1, 2, 3, 4, 5, 6.

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Now, can I multiply this?

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Well, what do we have?

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This is a 3 by 2 matrix.

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Can I multiply these two matrices?

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Well, what do we have to do?

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We get our row information from here, and our column

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information from here--

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Oh sorry, this isn't 2 by 1, this is 1 row, two columns.

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This is a 1 by 2, right?

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That's a 1 by 2 matrix.

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So can you multiply the 1 by 2 times a 3 by 2 matrix?

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So we get our row information from here, so we essentially

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have to multiply this by this times this column to get our

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first element, then this times this column to

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get our second element.

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And I don't know what happens from there, but let me-- well,

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can we multiply?

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Just the way we have defined our multiplication, or the dot

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product, can we multiply?

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Let's see, 2 times 1 plus 1 times 2.

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Then we don't have anything to do with the 3.

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So the way that we've defined matrix multiplication, you

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cannot multiply these two matrices.

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And you didn't have to go through that exercise.

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You could've looked at the dimensions, 1 by 2, and this

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is a 3 by 2.

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This 2 is not equal to this 3, the number of columns in the

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first are not equal to the number of rows in the second.

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So you can not multiply those two matrices.

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So that's something interesting to think about.

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And they're actually examples, and it's a good exercise for

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you to think about it, where you can multiply A times B,

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but you can't multiply B times A.

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So I want you think about examples where that happens.

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But anyway, I'm pushing 15 minutes, and I will see you in

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the next video.

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