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Dialogue: 0,0:00:05.99,0:00:09.47,Default,,0000,0000,0000,,Every square matrix has\Nassociated with it a special
Dialogue: 0,0:00:09.47,0:00:11.02,Default,,0000,0000,0000,,quantity called its determinant.
Dialogue: 0,0:00:11.64,0:00:16.09,Default,,0000,0000,0000,,And determinants turn out to be\Nvery useful when it comes to
Dialogue: 0,0:00:16.09,0:00:19.43,Default,,0000,0000,0000,,doing more advanced things with\Nmatrices like finding inverse
Dialogue: 0,0:00:19.43,0:00:22.03,Default,,0000,0000,0000,,matrices and solving\Nsimultaneous equations and later
Dialogue: 0,0:00:22.03,0:00:26.11,Default,,0000,0000,0000,,videos will cover these topics\Nwere going in this video. Just
Dialogue: 0,0:00:26.11,0:00:30.56,Default,,0000,0000,0000,,look at finding the determinant\Nof a two by two square matrix.
Dialogue: 0,0:00:30.56,0:00:32.79,Default,,0000,0000,0000,,So let's look at an example.
Dialogue: 0,0:00:35.99,0:00:37.66,Default,,0000,0000,0000,,Suppose we have a matrix A.
Dialogue: 0,0:00:38.39,0:00:42.78,Default,,0000,0000,0000,,Which is got entries\Nin it elements in it
Dialogue: 0,0:00:42.78,0:00:44.73,Default,,0000,0000,0000,,435 and minus one.
Dialogue: 0,0:00:46.24,0:00:49.54,Default,,0000,0000,0000,,So this is a two by two matrix.\NIt's a square matrix.
Dialogue: 0,0:00:50.24,0:00:54.26,Default,,0000,0000,0000,,Now the determinant is a single\Nvalue. It's a value that's
Dialogue: 0,0:00:54.26,0:00:57.90,Default,,0000,0000,0000,,generated by combining the\Nnumbers within the matrix in a
Dialogue: 0,0:00:57.90,0:01:01.56,Default,,0000,0000,0000,,special way for two by two\Nmatrices. The way isn't
Dialogue: 0,0:01:01.56,0:01:04.48,Default,,0000,0000,0000,,particularly hard, but equally\Nit's not particularly obvious
Dialogue: 0,0:01:04.48,0:01:09.58,Default,,0000,0000,0000,,that you would do it this way,\Nso we need to just see an
Dialogue: 0,0:01:09.58,0:01:14.33,Default,,0000,0000,0000,,example, so we want to find the\Ndeterminant of A and we normally
Dialogue: 0,0:01:14.33,0:01:18.71,Default,,0000,0000,0000,,denote that by the letters Det\Nbeing the first 3 letters of
Dialogue: 0,0:01:18.71,0:01:20.17,Default,,0000,0000,0000,,determinant. So we right.
Dialogue: 0,0:01:20.26,0:01:21.81,Default,,0000,0000,0000,,Debt of a.
Dialogue: 0,0:01:23.87,0:01:26.94,Default,,0000,0000,0000,,A moment working out\Ndeterminants instead of using
Dialogue: 0,0:01:26.94,0:01:30.40,Default,,0000,0000,0000,,round brackets around the\Nmatrix, we use vertical lines.
Dialogue: 0,0:01:32.00,0:01:35.64,Default,,0000,0000,0000,,But the numbers within the\Nvertical lines are exactly
Dialogue: 0,0:01:35.64,0:01:36.44,Default,,0000,0000,0000,,the same.
Dialogue: 0,0:01:38.09,0:01:40.67,Default,,0000,0000,0000,,And now we have to combine\Nthose numbers.
Dialogue: 0,0:01:41.71,0:01:45.64,Default,,0000,0000,0000,,The first thing we do is we look\Nat the numbers on the leading
Dialogue: 0,0:01:45.64,0:01:49.30,Default,,0000,0000,0000,,diagonal. That's the four and\Nthe minus one. And what we do is
Dialogue: 0,0:01:49.30,0:01:50.70,Default,,0000,0000,0000,,we find their product, we
Dialogue: 0,0:01:50.70,0:01:55.85,Default,,0000,0000,0000,,multiply them together. So 4 *\N-- 1 Now having dealt with the
Dialogue: 0,0:01:55.85,0:01:59.76,Default,,0000,0000,0000,,numbers on the leading diagonal,\Nwe've got 2 numbers left. The
Dialogue: 0,0:01:59.76,0:02:03.30,Default,,0000,0000,0000,,numbers that aren't on the\Nleading diagonal, and so we
Dialogue: 0,0:02:03.30,0:02:06.86,Default,,0000,0000,0000,,multiply those two numbers\Ntogether. So that's 3 * 5.
Dialogue: 0,0:02:08.11,0:02:11.08,Default,,0000,0000,0000,,So we've got the product of the\Nnumbers on the leading
Dialogue: 0,0:02:11.08,0:02:13.78,Default,,0000,0000,0000,,diagonal, and we've got the\Nproduct. The numbers that are
Dialogue: 0,0:02:13.78,0:02:16.75,Default,,0000,0000,0000,,not on the diagonal, and then\Nwe just take this second
Dialogue: 0,0:02:16.75,0:02:18.37,Default,,0000,0000,0000,,product away from the first\Nproduct.
Dialogue: 0,0:02:19.86,0:02:27.40,Default,,0000,0000,0000,,And then we work it out 4 * -- 1\Nis minus four. 3 * 5 is 15, so
Dialogue: 0,0:02:27.40,0:02:31.77,Default,,0000,0000,0000,,we take away 15 -- 4 takeaway.\N15 is minus 19.
Dialogue: 0,0:02:34.29,0:02:38.22,Default,,0000,0000,0000,,And so that's how we do every\Nsingle determinant for a two by
Dialogue: 0,0:02:38.22,0:02:41.84,Default,,0000,0000,0000,,two matrix. We simply find the\Nproduct of the numbers on the
Dialogue: 0,0:02:41.84,0:02:45.16,Default,,0000,0000,0000,,leading diagonal and take away\Nfrom that the product of the
Dialogue: 0,0:02:45.16,0:02:46.97,Default,,0000,0000,0000,,other two numbers. So let's do
Dialogue: 0,0:02:46.97,0:02:52.45,Default,,0000,0000,0000,,another example. Here's another\Ntwo by two matrix B with
Dialogue: 0,0:02:52.45,0:02:53.95,Default,,0000,0000,0000,,elements 6, two.
Dialogue: 0,0:02:55.33,0:02:57.18,Default,,0000,0000,0000,,Three and five.
Dialogue: 0,0:02:58.39,0:03:00.87,Default,,0000,0000,0000,,And to workout the\Ndeterminant of B.
Dialogue: 0,0:03:03.28,0:03:06.66,Default,,0000,0000,0000,,We find the product of the\Nnumbers on the leading diagonal
Dialogue: 0,0:03:06.66,0:03:10.96,Default,,0000,0000,0000,,is 6 and five. Then we take away\Nthe product of the two numbers.
Dialogue: 0,0:03:10.96,0:03:12.80,Default,,0000,0000,0000,,Those two numbers are two and
Dialogue: 0,0:03:12.80,0:03:16.57,Default,,0000,0000,0000,,three. So we're taking\Naway 2 * 3.
Dialogue: 0,0:03:17.80,0:03:24.47,Default,,0000,0000,0000,,So 6 * 5 is 32 *\N3 is 630 takeaway 6, and
Dialogue: 0,0:03:24.47,0:03:26.01,Default,,0000,0000,0000,,we're getting 24.
Dialogue: 0,0:03:27.68,0:03:31.64,Default,,0000,0000,0000,,The one more example here. So we\Ngot the matrix D.
Dialogue: 0,0:03:32.41,0:03:36.08,Default,,0000,0000,0000,,Which has got entries 64.
Dialogue: 0,0:03:36.64,0:03:43.13,Default,,0000,0000,0000,,Three and two. Now when we\Nworkout, the determinant of D.
Dialogue: 0,0:03:44.61,0:03:49.19,Default,,0000,0000,0000,,New the product, the numbers on\Nthe leading diagonal is 6 * 2.
Dialogue: 0,0:03:49.90,0:03:56.38,Default,,0000,0000,0000,,Then we take away the product of\Nthe two numbers, which is 3 * 4.
Dialogue: 0,0:03:56.38,0:04:03.72,Default,,0000,0000,0000,,So 6 * 2 is 12. Taking away 3\N* 4, she's also 12, and so we
Dialogue: 0,0:04:03.72,0:04:05.45,Default,,0000,0000,0000,,get the answer 0.
Dialogue: 0,0:04:06.75,0:04:10.11,Default,,0000,0000,0000,,So we see that every time we\Nworkout the determinant, what we
Dialogue: 0,0:04:10.11,0:04:11.51,Default,,0000,0000,0000,,get is a single number.
Dialogue: 0,0:04:12.21,0:04:18.31,Default,,0000,0000,0000,,Determinant of a is minus 19 the\Ndeterminant of B is 24
Dialogue: 0,0:04:18.31,0:04:20.85,Default,,0000,0000,0000,,determinant of D is 0.
Dialogue: 0,0:04:21.90,0:04:26.80,Default,,0000,0000,0000,,Now when a matrix has a zero\Ndeterminant, it has a especially
Dialogue: 0,0:04:26.80,0:04:31.69,Default,,0000,0000,0000,,given name to that property. We\Nsay that the matrix is singular.
Dialogue: 0,0:04:39.02,0:04:43.38,Default,,0000,0000,0000,,So any matrix which is singular\Nis a square matrix whose
Dialogue: 0,0:04:43.38,0:04:44.56,Default,,0000,0000,0000,,determinant is 0.
Dialogue: 0,0:04:45.75,0:04:49.89,Default,,0000,0000,0000,,These other matrices, where\Nthe determinant wasn't zero
Dialogue: 0,0:04:49.89,0:04:51.44,Default,,0000,0000,0000,,are called nonsingular.
Dialogue: 0,0:04:57.52,0:05:01.48,Default,,0000,0000,0000,,So it doesn't matter what the\Ndeterminant is, just so long as
Dialogue: 0,0:05:01.48,0:05:03.46,Default,,0000,0000,0000,,it isn't zero. That makes the
Dialogue: 0,0:05:03.46,0:05:08.00,Default,,0000,0000,0000,,matrix nonsingular. Finally,\Nwe'll just look at a general two
Dialogue: 0,0:05:08.00,0:05:13.67,Default,,0000,0000,0000,,by two matrix. So here we have a\Nmatrix A and the entries the
Dialogue: 0,0:05:13.67,0:05:15.70,Default,,0000,0000,0000,,elements in this matrix RABC&D.
Dialogue: 0,0:05:16.33,0:05:19.07,Default,,0000,0000,0000,,If you want to workout\Nthe determinant of a.
Dialogue: 0,0:05:22.67,0:05:27.13,Default,,0000,0000,0000,,We just follow the process we've\Nalready seen. We take the values
Dialogue: 0,0:05:27.13,0:05:30.48,Default,,0000,0000,0000,,that are on the leading\Ndiagonal. We multiply them
Dialogue: 0,0:05:30.48,0:05:35.32,Default,,0000,0000,0000,,together to get a D. We take the\Nother two values and multiply
Dialogue: 0,0:05:35.32,0:05:39.78,Default,,0000,0000,0000,,them together. It should be in C\Nand we subtract this second
Dialogue: 0,0:05:39.78,0:05:43.87,Default,,0000,0000,0000,,product away from the first\Nproduct, and so this is the
Dialogue: 0,0:05:43.87,0:05:47.97,Default,,0000,0000,0000,,general formula for any two by\Ntwo matrix. The elements are
Dialogue: 0,0:05:47.97,0:05:52.43,Default,,0000,0000,0000,,ABCD. The determinant is worked\Nout to be a * D minus.
Dialogue: 0,0:05:52.70,0:05:53.56,Default,,0000,0000,0000,,B * C.
Dialogue: 0,0:05:56.67,0:06:00.32,Default,,0000,0000,0000,,So that's all there is to know\Nabout 2, but the determinants of
Dialogue: 0,0:06:00.32,0:06:01.45,Default,,0000,0000,0000,,two by two matrices.