 What other properties of matrix arithmetic can we discover or prove? Well, how about distributivity? So I suppose I have some matrix P and some scalars A and B, both of which we'll assume are real numbers, and we want to prove, or possibly disprove, that AP plus BP is equal to A plus B times P. So notice that if this statement happens to be true, it's as if we took this matrix P on the right and distributed it among the scalars A and B, and so we might actually call this right distributivity. Well, if I want to show that two matrices are equal, I want to show that the corresponding entries are equal. So let's think about that. So for our matrix AP plus BP, there's two operations going on here. First, there's matrix addition, which I perform by adding the corresponding components, which means I need to add the entries of AP to the entries of BP. Since this is a scalar multiple of the matrix P, then those entries are going to be A times the entries of P. So I get A times some entry PIJ plus B times some entry PIJ. And that's going to be what the entries of the matrix AP plus BP are. Well, how about the entries of the matrix A plus B times P? And in this case, we have a scalar multiplication of the matrix P by the scalar A plus B. And so we'll multiply every entry of P by A plus B. And so that'll give us quantity A plus B times PIJ as the entries. And we take a look at our entries. The entries of the matrix AP plus BP are going to be A PIJ plus B PIJ. Meanwhile, the entries of the matrix A plus B times P are going to be A plus B times PIJ. And since these are equal, then the entries of the two matrices are equal. And so the matrices themselves are equal. And so we have proven that we have this right distributivity. And so we might take another lesson in life. If somebody takes the trouble to point out something called right distributivity, there's a good chance that they want to talk about, or at least mention, the idea of left distributivity. And so what happens if we have a factor on the left that we want to distribute over a sum? Because of the way we write things, what that's going to give us is sum scalar A times the sum of two matrices P plus Q. And if I have that factor of A on the left, the question at hand is whether or not it's true that A times the sum P plus Q is the same as AP plus AQ. Now we can show two matrices are equal by showing that their corresponding entries are equal. But there is one thing that's different about this problem than the previous problem. I can multiply any matrix by any scalar that I want to, and that's not an issue. However, I can't always add two matrices. And it's possible that this sum P plus Q, or AP plus AQ, is actually undefined. We might not actually be able to add the two matrices. Fortunately, we can invoke a rule of logic. Everything is true about nothing. For example, nothing is odd and even. And this is a true statement. Since we can't always add two matrices, we should consider what happens if we can't. If this sum AP plus AQ is undefined, it is nothing. And it's true that A times P plus Q is equal to AP plus AQ because this is a statement about nothing. Or another way of looking at it is that if we can't add the two matrices, then both sides are actually undefined. And so our statement is trivially true. So we only have to do work if we can do work. If we can add the two matrices, then they will be equal or not, depending on whether or not their components are equal. So let's examine them. So let's take a look at the entries of the matrix A times P plus Q. I can add the matrices P and Q together. And the entries of that sum will be Pij plus Qij. And this will be multiplied by the scalar A. Meanwhile, the entries of AP plus AQ will be A times the entry of P, Pij, plus A times the entry of Q, Qij. And so now we have the entries of A times P plus Q. We have the entries of AP plus AQ. And we can compare them. And we see that the entries are in fact equal. And so our matrices are equal. And if we put the two problems together, we've actually proven the following theorem on the distributivity of scalars over matrices. And so first, we prove that the quantity A plus B times P is AP plus VP. That's the distributivity of matrices over scalars. Or again, we could talk about this as right distributivity. And we also prove that A times quantity P plus Q is AP plus AQ. And here, we're distributing the scalar over the matrices. Or again, we could talk about this as left distributivity.