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Zero matrix & matrix multiplication

Just as any number multiplied by zero is zero, there is a zero matrix such that any matrix multiplied by it results in that zero matrix. Learn more from Sal. Created by Sal Khan.

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Video transcript

Voiceover: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 1 times any number and you get that number again. And you could view 1 as essentially the identity. The identity number, or this is the identity property of multiplication. You multiply 1 times any number, you get that number again. And that essentially inspired our thinking behind having identity matrices. Said hey, maybe there are some matrices that if I multiply times some other matrix, I'm 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 times an identity matrix and get that matrix again. Now if matrix A right over here is a square matrix, then in either situation, this identity matrix is going to be the same identity matrix. But if matrix A is not a square matrix, then these are going to be two different identity matrices, 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 spacial number in traditional multiplication, and that's a 0. So, we know that 0 times anything is equal to 0. Or, anything times 0 is equal to 0. So what would be the analogy if we're thinking about matrix multiplication? Well, it would be some matrix that if I were to multiply it times another matrix, I get, I guess you could say that same 0 matrix again. And that is what we call it. We call it a 0 matrix. So if I take some matrix A, and essentially, if I multiply it times one of these 0 matrices, or I multiply one of the 0 matrices times A, I should get another 0 matrix. And it depends on the dimensions. You might not get a 0 matrix with the same dimensions. It depends what the dimensions of A are going to be, but you could image what a 0 matrix might look like. For example, if A is 1,2,3,4, what's a 0 matrix that I could multiply this by to get another 0 matrix? Well, it might be pretty straight forward, if you just had a ton of zeros here, when you multiply this out, you're going to get this - you date 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 a - just to make the point clear - let's say we had a matrix 1,2,3,4,5,6. So over here, 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 columns as this one has rows, so 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 and I encourage you to multiply these two. 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 let me just write the dimensions. This is a 3x2 matrix, this is a 2x3matrix. So, we know that we have valid matrix multiplication going on right over same here. 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 3x3 matrix. So it's going to be a 3x3 matrix, and I'll leave it up to you to verify that all 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 everytime, you're multiplying say this row by this column to get that entry, we'll just have 0 times 1 plus 0 times 4 to get that 0 there. But the whole point of showing you this example is, we have one 0 matrix multiplying by this matrix right over here, and then we get another 0 matrix but it has different dimensions.