WEBVTT 00:00:05.990 --> 00:00:09.473 Every square matrix has associated with it a special 00:00:09.473 --> 00:00:11.021 quantity called its determinant. 00:00:11.640 --> 00:00:16.092 And determinants turn out to be very useful when it comes to 00:00:16.092 --> 00:00:19.431 doing more advanced things with matrices like finding inverse 00:00:19.431 --> 00:00:22.028 matrices and solving simultaneous equations and later 00:00:22.028 --> 00:00:26.109 videos will cover these topics were going in this video. Just 00:00:26.109 --> 00:00:30.561 look at finding the determinant of a two by two square matrix. 00:00:30.561 --> 00:00:32.787 So let's look at an example. 00:00:35.990 --> 00:00:37.658 Suppose we have a matrix A. 00:00:38.390 --> 00:00:42.782 Which is got entries in it elements in it 00:00:42.782 --> 00:00:44.734 435 and minus one. 00:00:46.240 --> 00:00:49.540 So this is a two by two matrix. It's a square matrix. 00:00:50.240 --> 00:00:54.255 Now the determinant is a single value. It's a value that's 00:00:54.255 --> 00:00:57.905 generated by combining the numbers within the matrix in a 00:00:57.905 --> 00:01:01.555 special way for two by two matrices. The way isn't 00:01:01.555 --> 00:01:04.475 particularly hard, but equally it's not particularly obvious 00:01:04.475 --> 00:01:09.585 that you would do it this way, so we need to just see an 00:01:09.585 --> 00:01:14.330 example, so we want to find the determinant of A and we normally 00:01:14.330 --> 00:01:18.710 denote that by the letters Det being the first 3 letters of 00:01:18.710 --> 00:01:20.170 determinant. So we right. 00:01:20.260 --> 00:01:21.808 Debt of a. 00:01:23.870 --> 00:01:26.942 A moment working out determinants instead of using 00:01:26.942 --> 00:01:30.398 round brackets around the matrix, we use vertical lines. 00:01:32.000 --> 00:01:35.636 But the numbers within the vertical lines are exactly 00:01:35.636 --> 00:01:36.444 the same. 00:01:38.090 --> 00:01:40.666 And now we have to combine those numbers. 00:01:41.710 --> 00:01:45.644 The first thing we do is we look at the numbers on the leading 00:01:45.644 --> 00:01:49.297 diagonal. That's the four and the minus one. And what we do is 00:01:49.297 --> 00:01:50.702 we find their product, we 00:01:50.702 --> 00:01:55.850 multiply them together. So 4 * -- 1 Now having dealt with the 00:01:55.850 --> 00:01:59.755 numbers on the leading diagonal, we've got 2 numbers left. The 00:01:59.755 --> 00:02:03.305 numbers that aren't on the leading diagonal, and so we 00:02:03.305 --> 00:02:06.855 multiply those two numbers together. So that's 3 * 5. 00:02:08.110 --> 00:02:11.080 So we've got the product of the numbers on the leading 00:02:11.080 --> 00:02:13.780 diagonal, and we've got the product. The numbers that are 00:02:13.780 --> 00:02:16.750 not on the diagonal, and then we just take this second 00:02:16.750 --> 00:02:18.370 product away from the first product. 00:02:19.860 --> 00:02:27.403 And then we work it out 4 * -- 1 is minus four. 3 * 5 is 15, so 00:02:27.403 --> 00:02:31.770 we take away 15 -- 4 takeaway. 15 is minus 19. 00:02:34.290 --> 00:02:38.216 And so that's how we do every single determinant for a two by 00:02:38.216 --> 00:02:41.840 two matrix. We simply find the product of the numbers on the 00:02:41.840 --> 00:02:45.162 leading diagonal and take away from that the product of the 00:02:45.162 --> 00:02:46.974 other two numbers. So let's do 00:02:46.974 --> 00:02:52.448 another example. Here's another two by two matrix B with 00:02:52.448 --> 00:02:53.951 elements 6, two. 00:02:55.330 --> 00:02:57.178 Three and five. 00:02:58.390 --> 00:03:00.868 And to workout the determinant of B. 00:03:03.280 --> 00:03:06.657 We find the product of the numbers on the leading diagonal 00:03:06.657 --> 00:03:10.955 is 6 and five. Then we take away the product of the two numbers. 00:03:10.955 --> 00:03:12.797 Those two numbers are two and 00:03:12.797 --> 00:03:16.574 three. So we're taking away 2 * 3. 00:03:17.800 --> 00:03:24.469 So 6 * 5 is 32 * 3 is 630 takeaway 6, and 00:03:24.469 --> 00:03:26.008 we're getting 24. 00:03:27.680 --> 00:03:31.640 The one more example here. So we got the matrix D. 00:03:32.410 --> 00:03:36.080 Which has got entries 64. 00:03:36.640 --> 00:03:43.130 Three and two. Now when we workout, the determinant of D. 00:03:44.610 --> 00:03:49.186 New the product, the numbers on the leading diagonal is 6 * 2. 00:03:49.900 --> 00:03:56.380 Then we take away the product of the two numbers, which is 3 * 4. 00:03:56.380 --> 00:04:03.724 So 6 * 2 is 12. Taking away 3 * 4, she's also 12, and so we 00:04:03.724 --> 00:04:05.452 get the answer 0. 00:04:06.750 --> 00:04:10.110 So we see that every time we workout the determinant, what we 00:04:10.110 --> 00:04:11.510 get is a single number. 00:04:12.210 --> 00:04:18.306 Determinant of a is minus 19 the determinant of B is 24 00:04:18.306 --> 00:04:20.846 determinant of D is 0. 00:04:21.900 --> 00:04:26.796 Now when a matrix has a zero determinant, it has a especially 00:04:26.796 --> 00:04:31.692 given name to that property. We say that the matrix is singular. 00:04:39.020 --> 00:04:43.376 So any matrix which is singular is a square matrix whose 00:04:43.376 --> 00:04:44.564 determinant is 0. 00:04:45.750 --> 00:04:49.886 These other matrices, where the determinant wasn't zero 00:04:49.886 --> 00:04:51.437 are called nonsingular. 00:04:57.520 --> 00:05:01.480 So it doesn't matter what the determinant is, just so long as 00:05:01.480 --> 00:05:03.460 it isn't zero. That makes the 00:05:03.460 --> 00:05:08.000 matrix nonsingular. Finally, we'll just look at a general two 00:05:08.000 --> 00:05:13.670 by two matrix. So here we have a matrix A and the entries the 00:05:13.670 --> 00:05:15.695 elements in this matrix RABC&D. 00:05:16.330 --> 00:05:19.066 If you want to workout the determinant of a. 00:05:22.670 --> 00:05:27.134 We just follow the process we've already seen. We take the values 00:05:27.134 --> 00:05:30.482 that are on the leading diagonal. We multiply them 00:05:30.482 --> 00:05:35.318 together to get a D. We take the other two values and multiply 00:05:35.318 --> 00:05:39.782 them together. It should be in C and we subtract this second 00:05:39.782 --> 00:05:43.874 product away from the first product, and so this is the 00:05:43.874 --> 00:05:47.966 general formula for any two by two matrix. The elements are 00:05:47.966 --> 00:05:52.430 ABCD. The determinant is worked out to be a * D minus. 00:05:52.700 --> 00:05:53.558 B * C. 00:05:56.670 --> 00:06:00.323 So that's all there is to know about 2, but the determinants of 00:06:00.323 --> 00:06:01.447 two by two matrices.