 Okay, welcome to this video. In this one, we're going to take a look at how to work out a determinant. What is it? How can you find determinants of varying sizes? So a determinant is a scalar. It's just a number. Could be positive, could be negative, could be zero. And it's derived from a square matrix, a single number derived from an entire matrix. Now the determinant of m would be written with m with the modulus signs either side of it, even though it can be a negative number. So here's an example of m and here is how we would write the determinant of m. Note that we don't bother writing straight sides and curved brackets as well. There's no point in that. It's just enough to have the straight line sides. So let's start with the definition of a 2 by 2 determinant. That's the easy case to look at. So let's write out a general 2 by 2 just using symbols. We'll have a, b, c, d. Written inside our straight line sides indicates a determinant. It's simply a, d minus b, c. Okay, so that's the falling diagonal. The leading diagonal is also called minus the rising diagonal, multiplied together. Very simple. Very simple. And that is how you can just look at and evaluate a 2 by 2 determinant. So for our example 1, 2, 3, 4. 1 times 4 is 4. Subtract of 2 times 3 is 6. And so that's going to give us minus 2 is the determinant. Okay, so a 3 by 3 determinant is going to be a bit more work. What we do is when we have a 3 by 3 determinant, we evaluate it by breaking it up into a number up to 3 smaller determinants, each of which is a 2 by 2. And for that we have our definition for immediate evaluation. So we break up bigger determinants into little ones and then evaluate them. Now I'm going to write out something here that's like a chessboard, but instead of black and white, I have pluses and minuses. You'll see why in a moment. The thing to notice though is that we alternate plus, minus, plus, minus along each row and each column in this 3 by 3 grid. Okay. So now let's work out a 3 by 3 determinant. Again, I will just use general symbols A, B, C, D, E, F, G, H, I. Right. Now first I have to choose a row or a column. I'm going to choose this top row for the first example. And I'm going to work along this row and I'm going to start with the A symbol. Now I go and I look on my chart and I see that there's a plus sign in that slot of my grid. That means I put down plus A. And now what I do is I ignore the whole row and the whole column that A is in. And I look at the remaining four numbers and I write a little determinant just made out of those guys in the same order they appear. So E, F is going to be in my main determinant there and H, I, those are the remaining four guys in the same order they appear. Now B, the next term, that has a minus sign according to my chart. So I will put in minus B and multiply it by, again, a smaller 2 by 2 determinant. The one I get if I delete the row and the column with B in it and look at the remaining guys, D, F, G, I. And I just write those guys out in the same order they appear as a small 2 by 2 determinant. Finally there's C. C appears with a plus sign according to my chart. So I need to put down plus C and I need to multiply by, well, we delete the row and column with C in it. And we just see the remaining determinant, D, E, G, H. So I simply imagine that that row and column was not there and then that's what the determinant becomes. And then of course those 2 by 2 determinants, I can just write down what they are using my rule of multiplying down the diagonal and subtracting the anti-diagonal. Okay, there we are. So that is, in general, what a 3 by 3 determinant evaluates to. But it's not the only way to do it. Let's write it out again. And this time choose, let's choose a column and a different one. Let's choose this column. I'm also allowed to work down this. So I would start with B as my first term and I delete the row and column with it in and I'd see what are the remaining terms and write them D, F, G, I. Except I've forgotten something, there's a minus sign attached to that particular entry. So that should actually have been minus B. Alright, and then similarly plus E and I delete the row and column which has E in it. And then I just make a 2 by 2 determinant from, in this case, it would be the corner elements A, C, G, I. And then finally minus H and delete the row and column with H in it. Make a 2 by 2 determinant of what's left, A, C, D, F. Okay, and of course I could then write out these 2 by 2 determinants explicitly. But the point is it will give me the same answer. Let's do an example and see why we would choose one method or the other. So here are just some random numbers I'm making up. Let's stick that in. It's 3 by 3. First off, let's work along the top row and as we did in our first example. So that's going to be 3. Let's put in the full determinant here and then minus 1. And again, the determinant I get by excluding the top row and middle column and then plus 2. That's going to be 7, 0, 5, minus 1. And I can go ahead and I can work out explicitly what this comes out at. Let's see what I'm doing here and in fact it will be 12 plus 20 minus 14 and it comes out as 18. So there we are. We've worked out a 3 by 3. But we could have done it in a different way. Let's say we went along this bottom row. That's fine. So then it will be 5 and I will be left with 1, 2, 0, 4 for my mini-terminant and the next element along a minus sign and it was a minus number anyway. Minus minus 1. That's going to be 3, 2, 7, 4. Let's just see how we've done that. 3, 2, 7, 4 by deleting the bottom row and middle column of that. Now what about the third element here? Well, we actually have a 0 plus 0 times some determinant. I don't even care what that is because it's been multiplied by 0. That's the beauty of it. So I've got 5 into 4 minus 0 and then we're going to have 4, 3s of 12 minus 14. So that's going to give us 20 minus 2 is 18. Same answer as before. Okay. What about if we have even bigger determinants than our 3 by 3 example there? If we go bigger still, for example, a 4 by 4, we're just going to break it up into a number of 3 by 3s and each of those would have to be broken up into 2 by 2s, lots of work. So here we are. Here's a general 4 by 4. We are going to expand it along a row or column. Let's say we want to expand it along this row, for example, and we'll take in turn A, B, C, D. And we'll need to know what sign to use. So here's our checkerboard or our chessboard pattern of pluses and minuses. Just extend it out now to a 4 by 4. And you can see the rule here is that if you like, if the row number plus the column number is an even number, then there's going to be a plus sign. And if it's odd, it's going to be a minus sign. You can confirm that for yourself. Look at this one. It's going to be at row 2 and column 3 and that's 5. And so that's a minus. That's one way to remember it or just draw it out. Anyway, we're going to use that rule. So we go ahead and we write plus A and now we need to do the entire 3 by 3 determinant that we get when we delete the row and column with A in it. So we just write out that little square block that we see is quite easy to copy across. And now we're going to have minus B and we need to delete the row and column and then transcribe across the elements that are left as a 3 by 3. Just being careful not to make any slips. And you see that we're going to continue this way. So let's delete this just to be completely explicit. I'll finish the job off. So I think I hope it's obvious what we're doing. We're onto plus C and now we're going to just have E F H I J L and M N P. And then finally minus D onto what we get if we delete the top row and rightmost column which is left over then E F G I J K M N O. So here we are. That's how we handle a 4 by 4. Each of these 3 by 3s would then have to be evaluated and so on. So a lot of work. And that's the end of the video.