Dimensions of identity matrix

Voiceover:Let's say that you've got some matrix C, trying my best to bold it, to make sure you realize that this is a matrix. Let's say that we know that it has a rows and b columns. It's an a by b matrix. Let's say that we are going to multiply it by some identity matrix. Once again let me do my best to attempt to bold this right over here. We're going to multiply the identity matrix I times C and of course we are going to get C again because that's the identity matrix, that's the property of the identity matrix. Of course C, we already know is an a by b matrix. A rows and b columns. Based on this, what are the dimensions of I going to be? I encourage you to pause this video and think about it on your own. We've already done this exercise a little bit, where we first looked at identity matrices but now we're doing it with a very ... We're multiplying the identity matrix times a very general matrix. I'm just even speaking in generalities about these dimensions. Well one thing we know is that matrix multiplication is only defined is if the column, the number of columns of the first matrix is equal to the number of rows of the second matrix. This one has a rows, so this one's going to have a columns. Now how many rows is this one going to have? We already know that matrix multiplication is only defined if the number of columns on the first matrix is equal to the number of rows on the second one. We know that the product gets its number of rows from the number of rows of the first matrix being multiplied. The product has a rows then the identity matrix right over here has to have a rows. What's interesting about this? When we first got introduced to identity matrices, we were multiplying, we picked out a three by three example and we got a three by three identity matrix. What's interesting about what we've just proven to ourselves is the identity matrix for any matrix, even a non square matrix, a and b could be two different values. The identity matrix for any matrix is going to be a square matrix. It's going to have the same number of rows and the same number of columns. When we think about identity matrices, we can really just say, well is this the identity matrix that is a four by four? Is it a three by three? Is it a two by two? Or I guess one by one? The convention is, it isn't even to write identity two by two is equal to one, zero, zero, one. The convention is actually just write I2 because you know it's going to be a two by two. It's going to be a two by two matrix, it's going to be one, zero, zero, one. Identity five is going to be a five by five matrix. It's going to be one, one, two, three, four. Zero, one, two, one, three. Zero, zero, one, zero, zero. Zero ... you get the idea, zero, zero, zero, one, zero. Zero, zero, zero, zero, one. Just like that. The whole point here is just to realize that your identity matrix is always going to be a square matrix and it works even when you're multiplying non square other matrices.