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

Video transcript

when you first learned multiplication many many many years ago you got exposed to the idea that one times I shouldn't use that symbol one times some number is equal to that number again that and that makes intuitive sense you're just losing one of this thing is just going to be that thing right over there and you could view it as one when you're thinking about regular multiplication or scalar multiplication it has this identity property so it has the identity property of multiplication one times some number is equal to that some number again since we're now exploring matrices and matrix multiplication the question arises well is there some matrix that has the same property for matrix multiplication so 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 and if I were to multiply it times any other matrix a that the resulting product is going to be is going to be matrix a again by the standard conventions of matrix multiplication and 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 our matrix a let's go three by three let's say it is one two three four five six seven eight nine what I encourage you to do is pause this video and try to think about whether you could construct some matrix a and first think about even what the dimensions of matrix I have to be in order to in order to when you multiply the two this way when you multiply I times a you get a again so I'm assuming you've given a go at it so let's think this through so let's throw matrix a down there so let's see copy and paste and let's first think about what the dimensions are going to have to be well when I multiply my matrix when I multiply my matrix I when I multiply my matrix I times a right over here I get a again I get a again so I'm multiplying something times a 3 by 3 3 by 3 matrix and I'm getting another 3 by 3 matrix so 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 well we already see that a has 3 rows so this character the identity matrix is going to have to have 3 columns so it's going to have to have 3 columns and we also know we also know that the dimensions of the product the rows 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 so let me so this is what defines this these middle two have to match and then the rows of the first matrix to find the rows of the product and then the columns of the second Matrix define the columns of the product so we know this has to be a 3 by 3 matrix now what else do we need do we know well we know what the product needs to be it also needs to be 1 2 3 4 5 6 7 8 9 so 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 so I'm gonna have to multiply something times 1 plus something else times 4 plus something else times 7 to get 1 well let's just think about in the most I guess you 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 this if we take this product this entry right over here is going to be 1 times 1 1 times 1 plus 0 times four zero times four plus 0 times 7 plus 0 times 7 so that worked out quite well but let's just make sure that that still holds what happens when we multiply this row x times this column right or times this column to get this entry right over here well it works out it's 1 times 2 plus 0 2 times 5 plus 0 times 8 so it makes sense you get 2 again and same thing when you do it for this third 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 well the second row right over here is going to determine what values we get over here so 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 and we want to have the 4 so one way to think well 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 and that works out for this next that works out for this next entry right over here 0 times 2 plus 1 times 5 plus 0 times 8 we get 5 and it'll work out the same for this entry over there now for this last entry for these 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 so to get the 7 we want to multiply this tie this row times this column or take the dot product of this row and that column so if we want the 7 let's multiply 0 times the 1 plus a 0 times the 4 plus a 1 times the 7 and just like that you'll see that at 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 and you take the dot product of that and that it gives you the 9 the 9 for that entry and so just like that we have constructed a 3 by 3 identity matrix so the 3 by 3 identity matrix is equal to 1 0 0 0 1 0 and 0 0 1 and as you will see whenever you construct an identity matrix if you're constructing a 2 by 2 identity matrix so I could 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 if you have a 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 ones down the diagonal going from the top left to the bottom right and 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 what about a times 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