Dimensions of identity matrix

let's say that you've got some matrix see trying my best to bold it to make sure you realize that this is a matrix and let's say that we know that it has a a rose and B columns so it's an a by B a by B matrix and let's say that we are going to multiply it by some identity matrix some identity matrix once again let me do my best attempt to bold this right over here so 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 entity matrix and of course C we already know is an a by B matrix a rows and B columns and B columns now what are going what based on this what are the dimensions of I going to be and I encourage you to pause this video and think about it on your own so 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 we don't even i'm just even speaking in generalities about its dimensions well one thing we know is that matrix multiplication is only defined is if the columns the number of columns of the first matrix is equal to the number of rows of the second Matrix so this one has a rows so this one's going to have a columns it's going to have a columns so it's going to have a columns now how many rows is this one going to have well 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 and we know that the product gets its number of rows we know the product gets its number of rows from the number of rows of the first matrix being multiplied so the product has a rows then the identity matrix right over here has two have a rose now what's interesting about this when we first got introduced to identity major 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 it's going to have the same number of rows and the same number of columns and so when we think about identity matrices we can really just say well you know is this is this the identity matrix that is a four by four is it a three by three is it a two by two or even a one by one and so you don't even so the convention isn't even to write identity 2 by 2 is equal to 1 0 0 1 the convention is actually just write I to because you know it's going to be a 2 by 2 it's a 2 well it's going to be a 2 by 2 matrix so it's going to be 1 0 0 1 identity 5 it's going to be a 5 by 5 matrix so it's going to be 1 1 2 3 4 0 1 2 1 2 3 0 0 1 1 0 0 0 you get the idea 0 0 0 1 0 0 0 0 0 1 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