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Well, after the last video, hopefully, we're a little

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familiar with how you add matrices.

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So now let's learn how to multiply matrices.

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And keep in mind, these are human-created definitions for

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matrix multiplication.

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We could have come up with completely different ways to

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multiply it.

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But I encourage you to learn this way because it'll help

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you in math class.

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And we will see later that there's actually a lot of

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applications that come out of this type of matrix

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multiplication.

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So let me think of two matrices.

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I will do two 2 by 2 matrices, and let's multiply them.

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Let's say-- let me pick some random numbers: 2,

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minus 3, 7, and 5.

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And I'm going to multiply that matrix, or that table of

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numbers, times 10, minus 8-- let me pick a good number

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here-- 12, and then minus 2.

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So now there might be a strong temptation-- and you know in

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some ways it's not even an illegitimate temptation-- to

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do the same thing with multiplication that we did

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with addition, to just multiply the corresponding

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terms. So you might be tempted to say, well, the first term

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right here, the 1, 1 term, or in the first row and first

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column, is going to be 2 times 10.

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And this term is going to be minus 3 times

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minus 8 and so forth.

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And that's how we added matrices so maybe it's a

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natural extension to multiply matrices the same way.

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And that is legitimate.

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One could define it that way, but that's not the way it is

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in the real world.

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And the way in the real world,

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unfortunately, is more complex.

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But if you look at a bunch of examples I

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think you'll get it.

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And you'll learn that it's actually fairly

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straightforward.

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So how do we do it?

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So this first term that's in the first row and its first

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column, is equal to essentially this first row's

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vector-- no, this first row vector--

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times this column vector.

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Now what do I mean by that, right?

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So it's getting it's row information from the first

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matrix's row, and it's getting it's column information from

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the second matrix's column.

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So how do I do that?

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If you're familiar with dot product, it's essentially the

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dot product of these two matrices.

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Or without saying it that fancy, it's just this: it's 2

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times 10, so 2-- I'm going to write small-- times 10, plus

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minus 3 times 12.

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I'm going to run out of space.

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And so what's this second term over here?

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Well, we're still on the first row of the product vector but

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now we're on the second column.

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We get our column information from here.

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So let's pick a good color-- this is a slightly different

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shade of purple.

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So now this is going to be-- I'll do that in another

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color-- 2 times minus 8-- let me just write out the number--

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2 times minus 8 is minus 16, plus minus 3 times minus 2--

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what's minus 3 times minus 2?

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That is plus 6, right?

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So that's in row 1 column 2.

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It's minus 16 plus 6.

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And then let's come down here.

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So now we're in the second row.

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So now we're going to use-- we're getting our row

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information from the first matrix-- I know this is

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confusing and I feel bad for you right now, but we're going

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to a bunch of examples and I think it'll make sense.

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So this term-- the bottom left term-- is going to be this row

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times this column.

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So it's going to be 7 times 10, so 70, plus 7 times 10

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plus 5 times 12, plus 60.

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And then the bottom right term is going to be 7 times minus

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8, which is minus 56 plus 5 times minus 2.

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So that's minus 10.

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So the final product is going to be 2 times 10 is 20, minus

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36, so that's minus 16 plus 6, that's 10.

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90-- was that what I said?

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No, it was-- 70, plus 60, that's 130.

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And then minus 56 minus 10, so minus 66.

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So there you have it.

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We just multiplied this matrix times this matrix.

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Let me do another example.

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And I think I'll actually squeeze it on this side so

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that we can write this side out a little bit more neatly.

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So let's take the matrix and now 1, 2, 3, 4, times the

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matrix 5, 6, 7, 8.

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Now we have much more space to work with so this should come

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out neater.

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OK, but I'm going to do the same thing, so to get this

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term right here-- the top left term-- we're going to take--

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or the one that has row 1 column 1-- we're going to take

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the row 1 information from here, and the column 1

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information from here.

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So you can view it as this row vector

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times this column vector.

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So it results, 1 times 5 plus 2 times 7.

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Right?

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There you go.

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And so this term, it'll be this row vector times this

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column vector-- let me do that in a different color-- will be

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1 times 6 plus 2 times 8.

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Let me write that down.

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So it's 1 times 6 plus 2 times 8.

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Now we go down to the second row.

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And we get our row information from the first vector-- let me

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circle it with this color-- and it is 3 times 5

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plus 4 times 7.

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And then we are in the bottom right, so we're in the bottom

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row and second column.

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So we get our row information from here and our column

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information from here.

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So it's 3 times 6 plus 4 times 8.

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And if we simplify, this is 5 plus--

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Well actually, let me just remind you where all the

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numbers came from.

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So we have that green color, right?

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This 1 and this 2, that's this 1 and this 2,

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this 1 and this 2.

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Right?

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And notice, these were in the first row and they're in the

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first row here.

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And this 5 and this 7?

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Well, that's this 5 and this 7, and this 5 and this 7.

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So, interesting.

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This was in column 1 of the second matrix and this is in

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column 1 in the product matrix.

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And similarly, the 6 and the 8.

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That's this 6, this 8, and then it's used here, this 6

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and this 8.

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And then finally this 3 and the 4 in the brown, so that's

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this 3, this 4, and this 3 and this 4.

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And we could of course simplify all of it.

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This was 1 times 5 plus 2 times 7, so that's 5 plus 14,

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so this is 19.

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This is 1 times 6 plus 2 times 8, so it's 6 plus

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16, so that's 22.

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This is 3 times 5 plus 4 times 7.

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So 15 plus 28, 38, 43-- if my math is correct-- and then we

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have 3 times 6 plus 4 times 8.

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So that's 18 plus 32, that's 50.

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So now let me ask you-- just so you know that the product

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matrix-- just write it neatly-- is

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19, 22, 43, and 50.

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So now let me ask you a question.

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When we did matrix addition we learned that if I had two

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matrices-- it didn't matter what order we added them in.

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So if we said, A plus B-- and these are matrices; that's why

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I'm making them all bold-- we said this is the same thing as

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B plus A, based on how we define matrix

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addition, B plus A.

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So now let me ask you a question.

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Is multiplying two matrices, is AB-- that's just means

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we're multiplying A and B-- is that the same thing as BA?

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Does it matter?

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Does the order of the matrix multiplication matter?

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And so, I'll tell you right now, it actually matters a

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tremendous amount.

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And actually there are certain matrices that you can add in

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one direction that you can't add in the other-- oh, that

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you can multiply in one way but you can't multiply in the

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other order.

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And well, I'll show you that in an example-- but just to

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show that this isn't even equal for most matrices, I

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encourage you to multiply these two matrices in the

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other order.

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Actually let me do that.

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Let me do that really fast just to prove

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the point to you.

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So let me delete all this top part.

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Let me delete all of it, and actually I can delete to this.

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So hopefully, you know that when I multiply this matrix

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times this matrix, I got this.

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So let me switch the order-- and I'll do it fairly fast

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just so as to not bore you-- so let me switch the order of

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the matrix multiplication.

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This is good as this is another example-- so I'm going

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to multiply this matrix: 5, 6, 7, 8, times this matrix-- and

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I just switched the order; and we're testing to see whether

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order matters-- 1, 2, 3, 4.

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Let's do it-- and I won't do all the colors and everything,

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I'll just do it systematically.

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I think you just have to see a lot of examples here-- So this

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first term gets its row information from the first

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matrix, column information from the second matrix.

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So it's 5 times 1 plus 6 times 3, so it's 5 times 1--

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Let me just write, actually edit.

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I'm going to skip a step here-- OK so it's 5 times 1

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plus 6 times 3, plus 18.

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What's the second term here?

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It's going to be 5 times 2 plus 6 times 4.

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So 5 times 2 is 10, plus 6 times 4 is 24.

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Right, now we just took this row times this

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column right here.

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OK now we're down here for the set-- so then we're doing this

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row, this element right here at the bottom left is going to

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use this row, and this column.

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So it's 7 times 1 plus 8 times 3.

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8 times 3 is 24.

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And then finally, to get this element we're essentially

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multiplying this row times this column, so it's 7 times 2

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is 14, plus 8 times 4, plus 32.

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So this is equal to 5 plus 18 is 23, 34.

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What's 7 plus 24?

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That's 31, 46.

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So notice, if we called this matrix A and this

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is matrix B, right?

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In the last example, we showed that A times B is equal to 19,

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22, 43, 50.

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And we just showed that, well, if you reverse the order, B

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times A is actually this completely different matrix.

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So the order in which you multiply

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matrices completely matters.

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So I'm actually running out of time.

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In the next video I going talk a little bit more about the

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types of matrix-- well, one, we know that order matters--

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and in the next video I'll show that what type of

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matrices can be multiplied by each other.

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When we added or subtracted matrices, we just said, well

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they have to have the same dimensions because you're

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adding or subtracting corresponding terms. But

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you'll see with multiplication it's a little bit different.

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And we'll do that in the next video.

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See you soon.

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