College Algebra: The Arithmetic of Matrices

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Well now we've seen the value of using matrices. At least you use the Gauss-Jordan method of solving equations and so forth, so you can see the power of just looking at a list of numbers, just having those numbers there and manipulating them somehow. Well now let's sort of abstract that to the general idea of a matrix, which is just this array or this list of numbers. And now, let's see how we can actually perform arithmetic on that whole list or that whole array at once.
So the first thing we could talk about is how do you take two matrices and add them or subtract them? Well, in fact, your intuition here would be probably just right on the mark. Suppose I have two matrices and I want to combine them. Say I want to add them. Well the first thing that is important to realize is that not all two matrices can be added together. They have to have the same dimensions, the same shape, in fact. For example, here you can see these two things can be added, because they're both two by three, two rows, three columns. But I could not, for example add that matrix to this matrix. That's just impossible. There's no way for me to do that, because they are just sort of different kinds of objects. There's no way to sort of put them together in a nice way. They sort of are fighting each other.
But this is nice and smooth and silky. You can't help but want to add them together. And in fact, the method for adding is probably exactly what you would guess. Namely, well the answer is going to be another two by three. Now I'm going to have to write the answer. I apologize for this, but unless you have those really big monitors, this is not going to fit on the screen. So I'm going to put the answer way up here. But you can see that in my box over here, in fact, everything lines up perfectly. Anyway, this is going to be another two by three. That's the answer. So a two by three plus a two by three gives me another two by three. And what do you think goes in the first spot? Well you just add the elements of the first spot. So that would be a 2 in this case. Here you'd add the elements in this spot, which would give you a 5. In the one-three position, you'd add those elements and see a 1. Here you'd add and see a 1. Here you'd add and see a 7. And here you'd add and see a 5. So you just add the corresponding terms. That's why you see it's difficult to add these two things together. There are no corresponding terms. There's nothing that corresponds to sort of one-three, because there's no third thing here. So in fact, you can only add matrices that look the same. And then you just add the corresponding terms and get something else.
Subtraction is, of course, the exact same deal. What would you do here? Well subtraction, not a problem, you would still get a, in this case, two by three matrix. And you subtract. 1 - 1 = 0, 2 - 3 = -1, -1 - 2 = -3, 0 - 1 = -1, 3 - 4 = -1, and 5 - 0 = 5. Adding and subtracting matrices, no problem at all, really easy, just make sure that the dimensions of the matrices that you're adding or subtracting are the same. In this case it's two by three. This is two by three. They can be combined.
That takes care of that. Up next we're going to take a look at multiplication, which will get a lot more interesting. I'll see you there.
Systems of Equations
Matrices
The Arithmetic of Matrices Page [1 of 1]