Defined matrix operations

so we have matrix D and matrix B and they ask us is DB defined is the product D times B defined so D times B is going to be defined as if let me let me make this very clear this is how I think about it so let me copy and paste it so I can do this on my scratch pad so to answer that question you know a scratch pad right over here let me paste the question right here so let's think about these two matrices let's think about these two matrices you first have matrix D you first have matrix D I'll do it some nice bold D here and it has three rows and three columns so it is a three by three matrix and then you want to multiply that times matrix B matrix B matrix B is a 2 by 2 matrix the only way that we know to define matrix multiplication is if these middle two numbers are the same if the number of columns D has is equal to the number of rows B has now in this case they clearly do not equal each other so matrix multiplication is not defined here so let's go back there and say no no DB is not defined let's do a few more of these examples so then we have a 2 by 1 you could view this as a 2 by 1 matrix or you could view this as a column vector this is another 2 by 1 matrix or a column vector C plus B defined well matrix addition is defined if both matrices have the exact same dimensions and these two matrices do have the exact same dimension so the reason why is because with matrix addition you just add every corresponding term so in the sum the top it'll actually be 4 plus 0 over negative 2 plus 0 which is still just going to be the same thing as this matrix up here but what they're asking is this defined absolutely these both are 2 by 1 matrices so yes it is defined let's do one more so once again they're asking us is the product a times e defined so here you have a 2 by 2 matrix let me copy and paste this just so we can I make sure that we can we know what we're talking about so get my scratch pad out so this top matrix right over here so matrix a matrix a is a 2 by 2 matrix and matrix e matrix e so we're going to multiply it times matrix e which has one row and two columns one row and two columns so in this scenario once again the number of row the number of sorry the number of columns matrix a has is two and the number of rows matrix e has is one so this will not be defined these two things have to be the same for them to be defined now what is interesting is if you did it the other way around if you took e if you took E times a if you take e times a let's check if this would have been defined matrix e is 1 by 2 1 row x 2 columns matrix a is 2 rows times 2 or 2 by is a 2 by 2 2 rows and 2 columns and so this would have been defined matrix e has two columns which is exactly the same number of rows that matrix a has and this really hits the point home that the order matters when you multiply matrices but for the sake of this question is a e defined no it isn't and so we can check our answer no it isn't