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Current time:0:00Total duration:7:59

Voiceover:When you first
learned multiplication many, many, many years ago, you got exposed to the
idea that 1 times ... I shouldn't use that symbol ... 1 times some number is
equal to that number again, and that makes intuitive sense. You're just literally
saying one of this thing is just going to be that
thing right over there. And you could view it as 1, when you're thinking about
regular multiplication or scalar multiplication, it has this identity property. It has the identity
property of multiplication. 1 times some number is equal
to that some number again. Since we're now exploring matrices and matrix multiplication, the question arises is there some matrix that has the same property
for matrix multiplication? To make that a little bit more concrete, is there some matrix I, and let me bold it as best
as I can in my handwriting, is there some matrix I that
if I were to multiply it times any other ... I think I over-bolded that one, but I'll just go with it. If I were to multiply it
times any other matrix, A, that the resulting product
is going to be matrix A again by the standard conventions
of matrix multiplication. To make that a little bit
concrete, let's just imagine. Let's just take an example for A. Let's say that our matrix A, let's go 3 by 3. Let's say it is 1, 2, 3, 4, 5, 6, 7, 8, 9. What I encourage you
to is pause this video and try to think about
whether you can construct some matrix I, and first think about
even what the dimensions of matrix I have to be in order to, when you multiply the two this way, when you multiply I times A, you get A again. I'm assuming you've given a go at it, so let's think this through. Let's throw matrix A down there. Let's say copy and paste. Let's first think about
what the dimensions are going to have to be. When I multiply my matrix I, when I multiply my matrix
I times A right over here, I get A again. I'm multiplying something times a 3 by 3, 3 by 3 matrix, and I'm getting another 3 by 3 matrix. There's a few things that we know. First of all, in order for
this matrix multiplication to even be defined, this matrix, the identity matrix, has to have the same number
of columns as A has rows. We already see that A has 3 rows, so this character, the identity matrix, is going to have to have 3 columns. It's going to have to have 3 columns. We also know that the
dimensions of the product, the rows of the product are defined by the rows of the first matrix, so this has to be also a 3 by 3, and of course, the columns of the product are defined by the columns
of the second matrix. This is what defines this. These middle two have to match, and then the rows of the first matrix define the rows of the product, and then the columns of the second matrix define the columns of the product. We know this has to be a 3 by 3 matrix. Now what else do we know? We know what the product needs to be. It also needs to be 1,
2, 3, 4, 5, 6, 7, 8, 9. Let's think about it. To get this first entry right over here, we're going to have to multiply this row, this row times this column, since you take the dot product of it. I'm going to have to
multiply something times 1 plus something else times 4 plus something else times 7 to get 1. Let's just think about it in the most, I guess we could say, naive possible way. What happens if we just
multiply 1 times this 1 to get 1 and then 0 times 4 and add to it and then 0 times 7. I think that works out. When you take this product, this entry right over here
is going to be 1 times 1, 1 times 1 plus 0 times 4, 0 times 4 plus 0 times 7, plus 0 times 7. That worked out quite well, but let's just make sure
that that still holds. What happens when we multiply
this row times this column or times this column to get
this entry right over here? It works out. It's 1 times 2 plus 0
times 5 plus 0 times 8, so it makes sense. You get 2 again. Same thing when you do
it for this 3rd column. 1 times 3 plus 0 times 6 plus
0 times 9 is going to be 3. Now what do we do in the second row? Let's think about it a little bit. The second row right over here is going to determine what
values we get over here. For example, to get this
entry right over there, we're going to multiply this row, we're going to multiply
this row times this column, times this column. We want it to have the 4, so one way to think about it, we just want this middle entry here, so let's multiply 0 times 1
plus 1 times 4 plus 0 times 7, and then we're going to get 4. That works out for this
next entry right over here. 0 times 2 plus 1 times 5 plus 0 times 8. We get 5. It will work out the same
for this entry over there. Now, for this last entry, for this bottom row right
over here of our product, to do that, we're going
to have to multiply this row times these columns, or take, I guess you could
say, the dot product. To get the 7, we want to multiply this
row times this column, or take the dot product of
this row and that column. If we want the 7, let's
multiply 0 times a 1 plus 0 times a 4 plus a 1 times the 7. Just like that, you'll
see that that works. That gives us a 7 for this entry. It gives us, when you take
the dot of this and that, it gives you an 8 for this entry. You take the dot product of that and that. It gives you the 9, the 9 for that entry. Just like that, we have constructed a 3 by 3 identity matrix. The 3 by 3 identity matrix
is equal to 1, 0, 0, 0, 1, 0, and 0, 0, 1. As you will see, whenever you construct an identity matrix, if you're constructing a
2 by 2 identity matrix, so I can say identity matrix 2 by 2, it's going to have a very similar pattern. It's going to be 1, 0, 0, 1. If you have a 4 by 4 identity matrix, it is going to be, you could guess it, 1, 0, 0, 0, 0, 1, 0, 0,
0, 0, 1, 0, 0, 0, 0, 1. You essentially just
have 1s down the diagonal going from the top left
to the bottom right. What's neat about identity matrices, you multiply it times any matrix, and you're going to get that matrix again. Now another thing I encourage you to do is we've just shown that
I times A is equal to A, but I'll let you do this after this video, what about A times I? We've seen that matrix multiplication, the order matters, so what happens here? If you take A times I, do you still get A?