3 x 3 determinant

In the last video we defined the notion of a determinant of a 2 by 2 matrix. So if I have some matrix-- let's just call it B-- if my matrix B looks like this, if its entries are a, b, c, d, we've defined to determinant of B. Which could also be written as B with these lines around it, which could also be written as the entries of the matrix with those lines around it, a, b, c, d. And I don't want you to get these confused. This is the matrix when you have the brackets. This is the determinant of the matrix, when you just have these straight lines. And this, by definition, was equal to ad minus bc. And you saw in the last video, or maybe you saw in the last video, what the motivation for this came from. When we figured out the inverse of B, we determined that it was equal to 1 over ad minus bc times another matrix, which was essentially these two entry swaps, you got a d and an a. And then these two entries made negative, so minus c and minus b. This was the inverse of b. And we said, well, when is this defined? This is defined as long as this character right here does not equal 0. So you said hey, this looks pretty important. Let's call this thing right there the determinant. And then we could say that B is invertible, if and only if, the determinant of B does not equal 0. Because if it equals 0, then this formula for your inverse won't be well defined. And we just got this from our technique of creating an augmented matrix whatnot. But the big take away is we defined this notion of a determinant it for a 2 by 2 matrix. Now the next question is, well that's just a 2 by 2, everything we do in linear algebra, we like to generalize it to higher numbers of rows and columns. So the next step, at least-- let's just do baby steps-- let's start with a 3 by 3. Let's define what its determinant is. So let me construct a 3 by 3 matrix here. Let's say my matrix A is equal to-- let me just write its entries-- first row, first column, first row, second column, first row, third column. Then you have a2 1, a2 2, a2 3. Then you have a3 1, third row first column, a3 2, and then a3 3. That is a 3 by 3 matrix. Three rows and three columns. This is 3 by 3. I am going to define the determinant of A. So this is a definition. I'm going to define the determinant of this 3 by 3 matrix A as being equal to-- and this is a little bit convoluted, but you'll get the hang of it eventually. In the next several videos we're just going to do a ton of determinants. So it just becomes a bit of second nature to you. It's a little computationally intensive sometimes. But it equals this first row. It equals a1 1 times the determinant of the matrix you get, if you get rid of this guy's column and row. So if you get rid of this guy's column and row, you're left with this matrix here. So times the determinant of the matrix a2 2, a2 3, a3 2, and then a3 3. Just like that. So that's our first entry and that's a plus this. And then I said it's a plus this, because the next entry's going to be a minus. You have a minus this guy right here. So then you're going to have minus a1 2 times the matrix you get if you eliminate his column and his row. So times, you're going to get these entries right there. So a2 1, a2 3, a3 1, and then you have a3 3. We're not quite done. You could probably guess with the next one's going to be. Then you're going to have a plus-- let me switch to a better color-- plus this guy. Plus a1 3 times the determinant of its-- I guess you could call it-- its sub-matrix. We'll call it that for now. So this matrix right here. So a2 1, a2 2, a3 1, a3 2. This is our definition of the determinant of a 3 by 3 matrix. And the motivation is, because when you take the determinant of a 3 by 3 it turns out-- I haven't shown it to you yet-- that the property is the same. That if the determinant of this is 0, you will not be able to find an inverse. And when I defined determinant in this way. If the determinant does not equal 0, you will be able to find an inverse. So that's where this came from. And I haven't shown you that yet. And I might not show you because it's super computational. It'll take a long time. It'll be very hairy and I'll make careless mistakes. But the motivation comes from the exact same place as the 2 by 2 version. But I think what you probably want to see right now is at least just this thing applied to an actual matrix, because this looks all abstract right now. But if we do it with an actual matrix, you'll actually see it's not too bad. So let's leave the definition up there, and let's say that I have the matrix 1, 2, 4, 2, 2, minus 1, 3, and 4, 0, 1. So by our definition of a determinant, the determinant of this guy right here-- so let's say I call that matrix C-- C is equal to that. So if we want to figure out the determinant of C, the determinant of C is equal to-- I take this guy right here, let me take that 1-- times the determinant of-- let's just call it the submatrix, right here. So we have a minus 1, we have a 3, we have a 0, and we have a 1. Just like that. Notice, I got rid of this guy's column and this guy's row. And I was just left with minus 1, 3, 0, 1. Next, I take this guy. And this is the trick. You have to alternate signs. If you start with a positive here, this next one's going to be a minus. So you're going to have minus 2 times the submatrix-- we can call it-- if we get rid of this guy's column and this guy's row. So 2, 3, 4, 1. I just blanked this out. If I could videotape my finger, I would cover my finger over this column right here and over that row, and you'd just see a 2, a 3, a 4, and a 1. And that's what I put right there. And then finally, we went plus, minus, plus. So finally, we'll have plus 4 times the determinant of the submatrix, if you get rid of that row in that column. So 2, minus 1, 4, 0. Now, these are pretty straightforward. These are not too bad to compute. Let's actually do it. So this is going to be equal to 1 times what? Minus 1 times 1. Let me just write it out. Minus 1 times 1, minus 0 times 3. This just comes from the definition of a 2 by 2 determinant. We've already defined that. And then we're going to have a minus 2 times 2 times 1, minus 4 times 3. And then finally, we're going to have a plus 4 times 2 times 0 minus minus 1 times 4. I wrote it all out so you can see. This thing right here is just this thing right here. And then you have the 4 out front. This thing right here was just this thing right here. So it's the determinant of the 2 by 2 submatrix for each of these guys. And if we compute this, this is equal to-- minus 1 times 1 is minus 1. Minus 0, that's 0. So this is a minus 1 times 1, so that's a minus 1. And then we get-- what is this equal to? This right here is 12. So you get 2 minus 12. Right? You get 2 times 1 minus 4 times 3. So it's minus 10. So that is equal to minus 10. And then you have a minus 10 times a minus 2. So that becomes a plus 20, right? Minus 2 times minus 10. And then finally, in the green, we have 2 times 0, that's just a 0. And then you have minus 1 times 4, which is minus 4. Then you have a minus sign here, so it's plus 4. So this all becomes a plus 4. Plus 4 times 4 is 16, so plus 16. And what do we get when we add this up? We get 20 plus 16 minus 1. It is equal to 35. We're done. We found the determinant of our 3 by 3 matrix. Not too bad. Right there, so that is equal to the determinant of C. So the fact that this isn't 0 tells you that C is invertible. In the next video, we'll try to extend this to n by n square matrices.