Properties of matrix multiplication
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 is if-- let me make this very clear. This is how I think about it. So let me copy and paste this so I can do this on my scratch pad. So to answer that question, get out the scratch pad right over here. Let me paste the question right here. So let's think about these two matrices. You first have matrix D. I'll do this a nice bold D here. And it has three rows and three columns. So it is a 3 by 3 matrix. And then you'll want to multiply that times 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 is 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. Is 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 dimensions. And 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 make sure that we know what we're talking about. So get my scratch pad out. So this top matrix right over here, so matrix A is a 2 by 2 matrix. And matrix E, so we're going to multiply it times matrix E, which has one row and two columns. So in this scenario once again, the number of rows-- 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 times A, let's check if this would have been defined. Matrix E is 1 by 2, one row times two columns. Matrix A is a 2 by 2, two rows and two 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 AE defined? No, it isn't. And so we can check our answer, no, it isn't.