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Current time:0:00Total duration:4:42

Zero matrix & matrix multiplication

Video transcript

we've been drawing analogies between I guess we could say traditional multiplication or scalar multiplication and the first one we drew is when you have traditional multiplication you multiply one times any number and you get that number again and you could view one is essentially the identity the identity number or this is the identity property of multiplication you multiply one times any number you get that number again and that essentially inspired our thinking behind having identity matrices that hey maybe there are some matrices that if I multiply it times some other matrix I am going to get that matrix again and you've probably shown for yourselves that you can do it in either way you could have some matrix x times an identity matrix and get that matrix again now if these if this if matrix a right over here is a square matrix then in either in in in either situation this identity matrix is going to be the same identity matrix but if matrix a is not a square matrix and these are going to be two different identity matrices dependent depending on the appropriate dimensions now let's see if we can extend this analogy between traditional multiplication and matrix multiplication we know that there's another special number in traditional multiplication and that's a zero so we know that zero times anything is equal to zero or anything times zero is equal to zero so what would be the analogy if we're thinking about matrix multiplication well would be some matrix that if I were to multiply it times another matrix I get I guess you could say that same zero matrix again and that is what we call it we call it a zero matrix so if I take some matrix a and essentially if I multiply it times one of these zero matrix or if I multiply one of the zero matrix matrices times a I should get a I should get another zero matrix and it depends on the dimensions you might not get a zero matrix with the same diamond it depends what the dimensions of a are going to be but you could imagine what a zero matrix might look like for example if a is 1 2 3 4 what's a zero matrix that I can multiply this by to get another zero matrix well it might be pretty straightforward if you just had a ton of zeros here when you multiply this out you're going to get this you take the dot product of this row and this column 0 times 1 plus 0 times 3 is going to be 0 you keep going 0 0 0 0 if we had if we had a and just to make the point clear let's say we had a matrix 1 2 3 4 5 6 5 6 so over here what's the we want to multiply this times let's see in order for the matrix multiplication to work my 0 matrix has got to have the same number of same number of columns as this one has rows so it's got to have it's got to have it's got to have 2 columns but I could make it have 3 rows so it could look like this 0 0 0 0 0 and I encourage you to multiply these to pause the video right now and see what you get well when you multiply them let's think about it so the top left entry so this is let me just write the dimensions this is a 3 by 2 matrix this is a 2 by 3 matrix so that we know that we have valid matrix multiplication going on right over here same number the number of columns in the first matrix is equal to the number of rows in the second one and we also know that the resulting product is going to be a 3 by 3 matrix so it's going to be a 3 by 3 matrix and I'll leave it up to you to verify that all the entries here are going to be all of the entries here are going to be 0 and it makes sense you could go through the math but you can see well you're just every time you're multiplying say this row by this column to get that entry we just have 0 times 1 plus 0 times 4 to get that zero there but the whole point of showing you this example is we have one zero matrix multiplying by this being multiplied by this matrix right over here and then we get another zero matrix but it has different dimensions