Transcription
Every square matrix n by n has an associated value called its determinant, shown by straight vertical brackets, such as that example there. Okay? All right. Looks like a matrix. The difference is the brackets aren't hooked, right? They're straight vertical brackets. If you see that, you see the straight vertical brackets, that's telling you I'm going to find the determinant of this matrix. Okay?
If you look at the examples below in that box, 2 x 2, there's also a "de" in front. What do you think the "de" stands for? Determinant. Determinate. Right. Okay. You'll see it both ways: either the straight vertical brackets or the "d" out in front of brackets. Either way, it's asking you to find the determinant. Okay? Which, um, again, the determinant is the value of a matrix. When it talks about the value, it lines up with the linear map of when you're solving using matrices to solve linear equations. Um, that's the association there. But to find the value of a matrix, to find the determinant, we're going to find the difference of the diagonals. We're going to find the difference of the diagonals. What are we finding, EJ? Difference of the uh diagonals. Diagonals. EJ, what are we finding? The difference diagonals. Difference of the diagonals. Good. All right.
So, if you look at this example here, you can only find the determinant again if it's a square matrix. Okay? It has to have the same dimensions, same rows, same columns. Otherwise, you can't find the determinant. Okay? So top left, difference of the diagonals. So we'll do 1 * 8 is 8 minus 4 * 5 is 20. So the determinant there would be -12. Yeah. Top right. Difference of the diagonals: 12 * 2 is 24 minus -1 * -1/2 is a positive half. So that answer would be [Music] 23.5. Bottom left. What do you think that determinant would be? Um, I think it's -22. Good. 18 minus 40 gives us Oh my goodness. 22. What? Why are we unvolving pi? It's not that big a deal. Yeah. 1/2 * 2π: 1/2 * 2π is just π. You can write 3.14 or π minus 4 * 1/8 is 1/2, which if you subtract that we'll say is about 2.64. Okay. So that would be the determinant of those matrices. All right.
So 3 x 3 matrix. All right. We're still finding the difference of diagonals, just a little bit longer process. Okay. So here, the first step when we're finding the determinant of a 3 x 3 matrix is we need to repeat the first two columns. So we need to rewrite 2, 5, and 10, and we need to rewrite -3, 1, and 3. Why don't we repeat the third one? Because it don't work like that. Why? Because when we do the diagonals, you'll see how it lines up. Okay. Um, so again, we're finding the difference of the diagonals. We just have three diagonals in each direction. We will start top left to bottom right. So first we'll have 2 * 1 * -1, which is -2 + 3 * -2 * 10 is -60 + 4 * 5 * 3 is 60. And then we'll have we'll find the difference of the diagonals going in the other direction. So we will do 10 * 1 * 4 is 40 + 3 * -2 * 2 is -12 + -1 * 5 * -3 is 15. So now add up each side, find the difference. So that'll be 118 - 43, should give us 75. Repeat the first two columns: 5, -3, 2, 6, 2, -3. We're finding the difference of what? We're finding the difference of what? Diagonals. Diagonals. Thank you. You're welcome. So, we're have 5 * 2 * 4 is 40 + 6 * 0 * 2 is 0 + -1 * -3 * -3 gives us a -9. Difference of the diagonals in the other direction: 2 * 2 * -1 is -4. And I got my lovely zero. -3 * 0 is 0. 0 * 5 is 0. And then 4 * -3 * 6 is -72. I'm going to add a plus sign in there in a second. I don't know why I didn't. So 40 - 9 gives me 31 minus a -76. So minus a negative makes that plus. So the determinant there would be 107. Condition. All right.
Next part: inverses of matrices. A matrix can have an inverse if and only if it is a square matrix. Some, but not all, square matrices have inverses. Okay. If uh if the determinant of a matrix is zero, it's not going to have a uh inverse. Okay. Then if we multiply a matrix by its inverse, the resulting matrix is an identity matrix, which is kind of like getting one. Okay? It's kind of like multiplying, getting one. But what an identity matrix looks like is it's a little pattern, right? A 2 x 2 would be 1 0 0 1. A 3 x 3 identity matrix would be 1 0 0 0 1 0 0 0 1. Okay? And that pattern would continue. A 4 x 4 would be 1 0 0 0 1 0 0 0 1 0 and 0 0 0 1. That pattern will just continue down. Excuse me. Okay.
So, we have that formula for the inverse of a matrix. You just have to memorize that. All right. It's uh the inverse of a matrix is one over the determinant. Then you flip A and D, and you make B and you multiply B and C by -1. Okay, one over the determinant. Flip A and D and multiply B and C by -1. Then you distribute one over the determinant and that would give you your inverse. So if you look at A here, okay, first step I need to get the determinant. How do I find the determinant? You do the diagonal. There it is. Difference of the diagonals. So 4 * 1 is 4 minus 3 * 2 is 6, gives us -2. So the inverse of that matrix, to set it up, the formula I'll have 1 over -2, which is -1/2. I will flip A and D, so that will be a 1 and that will be a 4 on the bottom right. And then I need to multiply B and C by -1. So that'll become a -3 and a -2. From there I will distribute one over the determinant. I will distribute that -1/2 to each term, and that'll give us -1/2, positive 3/2 or 1.5, 1, and -2. Okay, so if you multiplied that inverse by the original matrix, if you did rows by columns, what we just did, it would give you that identity matrix: 1 0 0 1. Okay, so that is what you would do to identify if matrices are inverses or not. You could multiply them, and if they give you an identity matrix, they would be inverses. Yes. Yes. AJ.
So if you look at B, first step is to find the determinant, right? 4 * 1/4 is 1 minus -3 * -3 is 9. What's 1 - 9? -8. It should be a positive one for the second number because it's 1/3 * -3, right? But we're finding the difference of the diagonals, right? So it'll be minus that positive one. So Oh, wait. Yeah. So that gives us -8 there, which means right that means there's an inverse. Okay. So the same for the next one. Nope. For the next one, the determinant: 3 * -2 is -6 and then it'll be minus -6 which gives us 0. So our determinant is 0. In our formula we'd write it as 1/0 because it's one over the determinant. We flip A and D. So we flip 3 and -2. And then we multiply B and C by -1. So that'll be a -2 and a -3. And then we distribute that -1/12 and we would have our inverse. Boom. Shakalaka.