[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:05.14,0:00:09.29,Default,,0000,0000,0000,,In previous videos, we've seen\Nhow to find the determinant of a
Dialogue: 0,0:00:09.29,0:00:13.10,Default,,0000,0000,0000,,two by two matrix. We've also\Nseen the role that the
Dialogue: 0,0:00:13.10,0:00:15.52,Default,,0000,0000,0000,,determinant plays in solving\Nsimultaneous linear equations,
Dialogue: 0,0:00:15.52,0:00:17.94,Default,,0000,0000,0000,,and in finding the inverse of a
Dialogue: 0,0:00:17.94,0:00:23.17,Default,,0000,0000,0000,,matrix. And what we're going to\Ndo in this video is we're going
Dialogue: 0,0:00:23.17,0:00:26.66,Default,,0000,0000,0000,,to look at three by three\Ndeterminant statements of square
Dialogue: 0,0:00:26.66,0:00:30.50,Default,,0000,0000,0000,,matrices with three rows and\Nthree columns. Now, if you can
Dialogue: 0,0:00:30.50,0:00:33.99,Default,,0000,0000,0000,,remember the rule for two by\Ntwo, determinant's was very
Dialogue: 0,0:00:33.99,0:00:36.43,Default,,0000,0000,0000,,straightforward, but perhaps not\Nvery intuitively obvious.
Dialogue: 0,0:00:37.17,0:00:41.10,Default,,0000,0000,0000,,With three by three matrices, it\Ngets a little bit more
Dialogue: 0,0:00:41.10,0:00:44.31,Default,,0000,0000,0000,,complicated, and, again not very\Nintuitively obvious and to
Dialogue: 0,0:00:44.31,0:00:48.24,Default,,0000,0000,0000,,workout the determinant of three\Nby three matrices, we need to
Dialogue: 0,0:00:48.24,0:00:51.81,Default,,0000,0000,0000,,first of all introduce some\Nother ideas, called minors and
Dialogue: 0,0:00:51.81,0:00:55.38,Default,,0000,0000,0000,,cofactors, and will start by\Ndoing that. And throughout this
Dialogue: 0,0:00:55.38,0:01:00.02,Default,,0000,0000,0000,,video will always use the same\Nmatrix A. So this is the matrix.
Dialogue: 0,0:01:00.02,0:01:05.02,Default,,0000,0000,0000,,Say that we're going to use and\Nyou can see that it's got three
Dialogue: 0,0:01:05.02,0:01:08.94,Default,,0000,0000,0000,,rows and three columns. We can\Nonly find determinants when we
Dialogue: 0,0:01:08.94,0:01:10.37,Default,,0000,0000,0000,,have a square matrix.
Dialogue: 0,0:01:10.44,0:01:14.06,Default,,0000,0000,0000,,The matrix that has the same\Nnumber of rows as it has
Dialogue: 0,0:01:14.06,0:01:19.31,Default,,0000,0000,0000,,columns. And to find the\Ndeterminant of this three by
Dialogue: 0,0:01:19.31,0:01:24.57,Default,,0000,0000,0000,,three matrix, we first of all\Nneed to understand what's meant
Dialogue: 0,0:01:24.57,0:01:30.78,Default,,0000,0000,0000,,by a minor. Every element within\Nour matrix has its own minor, so
Dialogue: 0,0:01:30.78,0:01:36.100,Default,,0000,0000,0000,,we spell minor like this MINOR,\Nand the minor of an element is
Dialogue: 0,0:01:36.100,0:01:43.21,Default,,0000,0000,0000,,the value of the determinant we\Nget when we cross out the row
Dialogue: 0,0:01:43.21,0:01:45.12,Default,,0000,0000,0000,,and the column containing.
Dialogue: 0,0:01:45.14,0:01:49.16,Default,,0000,0000,0000,,The element we're looking at, so\Nlet's look at this element 7.
Dialogue: 0,0:01:49.66,0:01:54.88,Default,,0000,0000,0000,,Element 7 is in the first row\Nand the first column. So if we
Dialogue: 0,0:01:54.88,0:01:57.12,Default,,0000,0000,0000,,cross out that throw in that
Dialogue: 0,0:01:57.12,0:02:00.37,Default,,0000,0000,0000,,column. I'm going to hide them\Nrather than across the Mount.
Dialogue: 0,0:02:02.59,0:02:06.97,Default,,0000,0000,0000,,You see that we are left with a\N2 by two matrix and we can find
Dialogue: 0,0:02:06.97,0:02:08.62,Default,,0000,0000,0000,,the determinant of that two by
Dialogue: 0,0:02:08.62,0:02:15.65,Default,,0000,0000,0000,,two matrix. So the minor of\Nthe element 7 in Matrix A is
Dialogue: 0,0:02:15.65,0:02:21.96,Default,,0000,0000,0000,,this two by two determinant\Nwith entries 3 -- 1 four and
Dialogue: 0,0:02:21.96,0:02:23.01,Default,,0000,0000,0000,,minus 2.
Dialogue: 0,0:02:24.12,0:02:27.59,Default,,0000,0000,0000,,Remember the rule for\Nevaluating a two by two
Dialogue: 0,0:02:27.59,0:02:31.07,Default,,0000,0000,0000,,determinant is to multiply\Nthe two elements on the
Dialogue: 0,0:02:31.07,0:02:34.54,Default,,0000,0000,0000,,leading diagonal and then\Nsubtract the product of the
Dialogue: 0,0:02:34.54,0:02:38.40,Default,,0000,0000,0000,,two elements that aren't on\Nthe leading diagonal. So we
Dialogue: 0,0:02:38.40,0:02:40.33,Default,,0000,0000,0000,,do 3 * -- 2.
Dialogue: 0,0:02:42.13,0:02:45.42,Default,,0000,0000,0000,,And then we take away 4 * -- 1.
Dialogue: 0,0:02:46.48,0:02:51.64,Default,,0000,0000,0000,,So that gives us minus 6\N-- -- 4.
Dialogue: 0,0:02:52.76,0:02:56.78,Default,,0000,0000,0000,,She's minus 6 + 4, which gives\Nus minus 2.
Dialogue: 0,0:02:57.46,0:03:03.45,Default,,0000,0000,0000,,So the minor of our elements 7\Nfrom our original three by three
Dialogue: 0,0:03:03.45,0:03:06.22,Default,,0000,0000,0000,,Matrix is the value minus 2.
Dialogue: 0,0:03:08.26,0:03:13.62,Default,,0000,0000,0000,,Now we can do the same thing for\Nany element in the matrix and
Dialogue: 0,0:03:13.62,0:03:18.22,Default,,0000,0000,0000,,find its minor. So let's pick\Nthis four. The four that's in
Dialogue: 0,0:03:18.22,0:03:22.81,Default,,0000,0000,0000,,the 3rd row and the second\Ncolumn, and we're going to find
Dialogue: 0,0:03:22.81,0:03:24.35,Default,,0000,0000,0000,,the minor of four.
Dialogue: 0,0:03:27.00,0:03:33.65,Default,,0000,0000,0000,,So we go\Nthrough the same
Dialogue: 0,0:03:33.65,0:03:39.56,Default,,0000,0000,0000,,process. Before is in the bottom\Nrow, so we cross that out and
Dialogue: 0,0:03:39.56,0:03:43.09,Default,,0000,0000,0000,,it's in the second column. So we\Ncross that out.
Dialogue: 0,0:03:44.15,0:03:47.02,Default,,0000,0000,0000,,And so we see we're left\Nwith a 2 by two matrix.
Dialogue: 0,0:03:48.23,0:03:51.09,Default,,0000,0000,0000,,And so we workout the\Ndeterminant of that two by
Dialogue: 0,0:03:51.09,0:03:51.66,Default,,0000,0000,0000,,two matrix.
Dialogue: 0,0:03:53.71,0:03:57.45,Default,,0000,0000,0000,,These are two by two Matrix 7,\None, 0 -- 1.
Dialogue: 0,0:03:58.00,0:04:03.95,Default,,0000,0000,0000,,So we do 7 * -- 1, which is\Nminus 7 and then we take away
Dialogue: 0,0:04:03.95,0:04:07.67,Default,,0000,0000,0000,,nought times one we're taking\Naway nought. And so we're
Dialogue: 0,0:04:07.67,0:04:09.90,Default,,0000,0000,0000,,getting an answer of minus 7.
Dialogue: 0,0:04:10.66,0:04:15.54,Default,,0000,0000,0000,,So the minor of the element for\Nin this matrix is minus 7.
Dialogue: 0,0:04:19.57,0:04:24.21,Default,,0000,0000,0000,,Now we could, but I'm not going\Nto now. We could workout the
Dialogue: 0,0:04:24.21,0:04:29.21,Default,,0000,0000,0000,,minor of every single element in\Nour matrix A and we get a value
Dialogue: 0,0:04:29.21,0:04:33.85,Default,,0000,0000,0000,,of the minor. So we get 9\Nvalues, one for each element of
Dialogue: 0,0:04:33.85,0:04:34.92,Default,,0000,0000,0000,,the matrix A.
Dialogue: 0,0:04:35.46,0:04:38.89,Default,,0000,0000,0000,,Now you can see now what the\Nprocess is to work those apps.
Dialogue: 0,0:04:39.51,0:04:43.25,Default,,0000,0000,0000,,I want to go now from minus to
Dialogue: 0,0:04:43.25,0:04:46.21,Default,,0000,0000,0000,,cofactors. And to get\Nfrom minors cofactors,
Dialogue: 0,0:04:46.21,0:04:49.50,Default,,0000,0000,0000,,I need the idea of what\Nwe call a place sign.
Dialogue: 0,0:04:50.74,0:04:54.79,Default,,0000,0000,0000,,Every place within the Matrix\Nhas a sign associated with it.
Dialogue: 0,0:04:55.80,0:04:59.83,Default,,0000,0000,0000,,We always start with a plus in\Nthe top left hand corner and
Dialogue: 0,0:04:59.83,0:05:03.86,Default,,0000,0000,0000,,then we alternate the signs. So\Nas we go across the road, we're
Dialogue: 0,0:05:03.86,0:05:05.72,Default,,0000,0000,0000,,going to go plus minus plus.
Dialogue: 0,0:05:07.06,0:05:12.87,Default,,0000,0000,0000,,As we go down the column, we're\Ngoing to go plus, minus plus.
Dialogue: 0,0:05:14.04,0:05:18.25,Default,,0000,0000,0000,,You can see that if we fill this\Nin, it will look like this.
Dialogue: 0,0:05:19.05,0:05:23.86,Default,,0000,0000,0000,,This is the matrix of place\Nsigns. So for instance, the
Dialogue: 0,0:05:23.86,0:05:28.66,Default,,0000,0000,0000,,element 4 we've looked at\Nbefore has a place sign of
Dialogue: 0,0:05:28.66,0:05:33.47,Default,,0000,0000,0000,,minus, whereas the seven that\Nwe've looked at as a place
Dialogue: 0,0:05:33.47,0:05:38.72,Default,,0000,0000,0000,,sign of plus and we used to\Nplay signs to convert minors
Dialogue: 0,0:05:38.72,0:05:40.90,Default,,0000,0000,0000,,into what we call cofactors.
Dialogue: 0,0:05:42.40,0:05:50.22,Default,,0000,0000,0000,,So the cofactor of the element\N7 is equal to the place
Dialogue: 0,0:05:50.22,0:05:53.17,Default,,0000,0000,0000,,sign. Times
Dialogue: 0,0:05:53.17,0:05:58.79,Default,,0000,0000,0000,,the minor.\NThat's true for all the
Dialogue: 0,0:05:58.79,0:06:03.65,Default,,0000,0000,0000,,elements. Every element has a\Ncofactor which we find by doing
Dialogue: 0,0:06:03.65,0:06:08.21,Default,,0000,0000,0000,,the product of its place sign\Nwith the value of its minor.
Dialogue: 0,0:06:08.72,0:06:12.91,Default,,0000,0000,0000,,So if the element 7, the place\Nsign was plus.
Dialogue: 0,0:06:13.53,0:06:19.07,Default,,0000,0000,0000,,And then at 7 the minor was\Nminus two. So we get plus minus
Dialogue: 0,0:06:19.07,0:06:21.45,Default,,0000,0000,0000,,two, which is just equal to
Dialogue: 0,0:06:21.45,0:06:28.11,Default,,0000,0000,0000,,minus 2. To the cofactor of the\Nelement 7 in this matrix A is
Dialogue: 0,0:06:28.11,0:06:33.95,Default,,0000,0000,0000,,minus 2. If you take the\Nelement 4, the cofactor.
Dialogue: 0,0:06:34.53,0:06:40.95,Default,,0000,0000,0000,,A full Is equal to its place\Nsign times it's minor, so the
Dialogue: 0,0:06:40.95,0:06:46.22,Default,,0000,0000,0000,,minor of four we saw was minus\Nseven. The place sign is a
Dialogue: 0,0:06:46.22,0:06:51.10,Default,,0000,0000,0000,,minus, so we don't play sign\Nminus times the value of the
Dialogue: 0,0:06:51.10,0:06:55.56,Default,,0000,0000,0000,,minor minus seven, so minus\Nminus seven is plus seven. So
Dialogue: 0,0:06:55.56,0:07:00.84,Default,,0000,0000,0000,,the cofactor of Element 4 in\Nthis matrix is 7 when they are
Dialogue: 0,0:07:00.84,0:07:04.90,Default,,0000,0000,0000,,going to calculate the\Ndeterminant of our matrix A by
Dialogue: 0,0:07:04.90,0:07:06.12,Default,,0000,0000,0000,,using the cofactors.
Dialogue: 0,0:07:06.65,0:07:10.32,Default,,0000,0000,0000,,So if we look at what we've\Nwritten down here, what I've
Dialogue: 0,0:07:10.32,0:07:14.61,Default,,0000,0000,0000,,done is I've taken our matrix A\Nand I've worked out the 9 minus.
Dialogue: 0,0:07:14.61,0:07:18.28,Default,,0000,0000,0000,,So the minor of seven. We've\Nalready seen his minus two and
Dialogue: 0,0:07:18.28,0:07:22.26,Default,,0000,0000,0000,,the minor of four we've seen as\Nminus seven. I've worked out all
Dialogue: 0,0:07:22.26,0:07:25.93,Default,,0000,0000,0000,,the other minors and I've put\Nthem into this matrix here that
Dialogue: 0,0:07:25.93,0:07:30.13,Default,,0000,0000,0000,,I've called M. Remember that we\Nhave a place sign matrix that
Dialogue: 0,0:07:30.13,0:07:33.71,Default,,0000,0000,0000,,starts with a plus in the top\Nleft hand corner and alternate
Dialogue: 0,0:07:33.71,0:07:35.79,Default,,0000,0000,0000,,sign as you go down or across.
Dialogue: 0,0:07:37.01,0:07:40.81,Default,,0000,0000,0000,,And then we calculate the\Ncofactors by taking the minor
Dialogue: 0,0:07:40.81,0:07:44.61,Default,,0000,0000,0000,,and multiplying by its place\Nsite. So I've assembled into
Dialogue: 0,0:07:44.61,0:07:48.79,Default,,0000,0000,0000,,the Matrix, see the cofactors\Nof all the elements of our
Dialogue: 0,0:07:48.79,0:07:53.73,Default,,0000,0000,0000,,matrix A and you can see that\Nyou get the cofactors from the
Dialogue: 0,0:07:53.73,0:07:56.77,Default,,0000,0000,0000,,minors simply by multiplying\Nby the place site.
Dialogue: 0,0:07:57.96,0:07:59.76,Default,,0000,0000,0000,,Now to workout the determinant
Dialogue: 0,0:07:59.76,0:08:04.72,Default,,0000,0000,0000,,of a. What we have to do is we\Ncan pick any row or any column.
Dialogue: 0,0:08:05.77,0:08:09.52,Default,,0000,0000,0000,,So let's say we were to pick the\Nfirst row and you see that the
Dialogue: 0,0:08:09.52,0:08:12.52,Default,,0000,0000,0000,,row has three elements in it\Nwould pick the column that would
Dialogue: 0,0:08:12.52,0:08:14.27,Default,,0000,0000,0000,,have had three elements in it as
Dialogue: 0,0:08:14.27,0:08:19.10,Default,,0000,0000,0000,,well. What we do is we pick\Nour row. We've got three
Dialogue: 0,0:08:19.10,0:08:21.88,Default,,0000,0000,0000,,elements and they've each got\Ntheir own cofactor.
Dialogue: 0,0:08:23.02,0:08:27.15,Default,,0000,0000,0000,,So we're going to do is going to\Nform the product of element
Dialogue: 0,0:08:27.15,0:08:31.61,Default,,0000,0000,0000,,times its cofactor, so we'll be\Ndoing 7 * -- 2 two times, three
Dialogue: 0,0:08:31.61,0:08:32.88,Default,,0000,0000,0000,,under 1 * 9.
Dialogue: 0,0:08:33.92,0:08:36.60,Default,,0000,0000,0000,,I've just explained what the\Nsteps are for calculating
Dialogue: 0,0:08:36.60,0:08:38.09,Default,,0000,0000,0000,,determinant, so let's now write
Dialogue: 0,0:08:38.09,0:08:42.10,Default,,0000,0000,0000,,it all down. We've seen that the\Nonly things that really matter
Dialogue: 0,0:08:42.10,0:08:45.50,Default,,0000,0000,0000,,are the matrix we started with\Nand its cofactors, so that's all
Dialogue: 0,0:08:45.50,0:08:47.21,Default,,0000,0000,0000,,we've got on on the sheet.
Dialogue: 0,0:08:48.38,0:08:49.88,Default,,0000,0000,0000,,So here's our matrix A.
Dialogue: 0,0:08:50.80,0:08:52.92,Default,,0000,0000,0000,,And here's the matrix of
Dialogue: 0,0:08:52.92,0:08:56.71,Default,,0000,0000,0000,,cofactors. Remember what we need\Nto do is we were picking the
Dialogue: 0,0:08:56.71,0:08:59.97,Default,,0000,0000,0000,,first row. We could pick any row\Nor column, but we've chosen to
Dialogue: 0,0:08:59.97,0:09:00.98,Default,,0000,0000,0000,,pick the first row.
Dialogue: 0,0:09:01.57,0:09:03.10,Default,,0000,0000,0000,,And workout the determinant of
Dialogue: 0,0:09:03.10,0:09:10.38,Default,,0000,0000,0000,,a. Or write Det the determinant,\Nthe T brackets a meaning of the
Dialogue: 0,0:09:10.38,0:09:11.99,Default,,0000,0000,0000,,determinant of a.
Dialogue: 0,0:09:12.52,0:09:16.08,Default,,0000,0000,0000,,And sometimes you'll see this\Nwritten with vertical lines.
Dialogue: 0,0:09:16.59,0:09:19.26,Default,,0000,0000,0000,,To mean the determinant of a.
Dialogue: 0,0:09:20.53,0:09:25.75,Default,,0000,0000,0000,,And what we do, because we've\Nchosen the first row, we write
Dialogue: 0,0:09:25.75,0:09:30.54,Default,,0000,0000,0000,,down the elements of the row\N7211, and we multiply each
Dialogue: 0,0:09:30.54,0:09:34.88,Default,,0000,0000,0000,,element by its cofactor. So\Nintroduce 7 * -- 2.
Dialogue: 0,0:09:36.24,0:09:38.49,Default,,0000,0000,0000,,We're going to do 2 * 3.
Dialogue: 0,0:09:40.13,0:09:42.47,Default,,0000,0000,0000,,And we're going to do 1 * 9.
Dialogue: 0,0:09:43.12,0:09:47.61,Default,,0000,0000,0000,,So we can get these three\Nproducts and then we add them
Dialogue: 0,0:09:47.61,0:09:54.79,Default,,0000,0000,0000,,up. So 7 * -- 2 is minus\N14. Two times three is 6 of 1
Dialogue: 0,0:09:54.79,0:10:02.02,Default,,0000,0000,0000,,* 9 is 9, so we workout minus\N14 + 6 + 9. We get the
Dialogue: 0,0:10:02.02,0:10:07.41,Default,,0000,0000,0000,,answer one. So the determinant\Nof a is equal to 1.
Dialogue: 0,0:10:08.37,0:10:11.54,Default,,0000,0000,0000,,Now you're probably thinking,\Nwell, what if I chosen a
Dialogue: 0,0:10:11.54,0:10:15.03,Default,,0000,0000,0000,,different row or a different\Ncolumn? Surely I would have got
Dialogue: 0,0:10:15.03,0:10:18.20,Default,,0000,0000,0000,,a different answer, so let's\Nchoose a different row, a
Dialogue: 0,0:10:18.20,0:10:21.05,Default,,0000,0000,0000,,different column, see what\Nhappens. Let's choose the second
Dialogue: 0,0:10:21.05,0:10:27.31,Default,,0000,0000,0000,,column. So if we're going to use\Nthe second column to workout the
Dialogue: 0,0:10:27.31,0:10:31.96,Default,,0000,0000,0000,,determinant of a, we've got\Nelements 2, three and four, 2,
Dialogue: 0,0:10:31.96,0:10:33.23,Default,,0000,0000,0000,,three, and four.
Dialogue: 0,0:10:33.77,0:10:38.61,Default,,0000,0000,0000,,And we multiply these by their\Ncofactors. 2 gets multiplied by
Dialogue: 0,0:10:38.61,0:10:43.01,Default,,0000,0000,0000,,three, 3 gets multiplied by\Nminus 11, and four gets
Dialogue: 0,0:10:43.01,0:10:44.33,Default,,0000,0000,0000,,multiplied by 7.
Dialogue: 0,0:10:45.27,0:10:46.92,Default,,0000,0000,0000,,We got the three products.
Dialogue: 0,0:10:47.48,0:10:48.67,Default,,0000,0000,0000,,And we add them up.
Dialogue: 0,0:10:49.84,0:10:57.24,Default,,0000,0000,0000,,So 2 * 3 is six\N3 * -- 11 is minus
Dialogue: 0,0:10:57.24,0:11:00.95,Default,,0000,0000,0000,,33 and 4 * 7 is
Dialogue: 0,0:11:00.95,0:11:04.50,Default,,0000,0000,0000,,28. Work this out. We get one.
Dialogue: 0,0:11:05.05,0:11:08.41,Default,,0000,0000,0000,,So we get the same value.
Dialogue: 0,0:11:09.19,0:11:13.26,Default,,0000,0000,0000,,And we will always get the same\Nvalue whichever row or column
Dialogue: 0,0:11:13.26,0:11:14.95,Default,,0000,0000,0000,,that we choose to use.
Dialogue: 0,0:11:15.48,0:11:19.92,Default,,0000,0000,0000,,Doesn't matter, it's completely\Nour choice. So what this means
Dialogue: 0,0:11:19.92,0:11:25.25,Default,,0000,0000,0000,,is that if we're working out the\Ndeterminant of a matrix, we
Dialogue: 0,0:11:25.25,0:11:28.80,Default,,0000,0000,0000,,actually don't need to workout\Nall nine cofactors.
Dialogue: 0,0:11:30.36,0:11:35.42,Default,,0000,0000,0000,,You see, when we use the first\Nrow, we only needed these three
Dialogue: 0,0:11:35.42,0:11:39.31,Default,,0000,0000,0000,,cofactors. When we use the\Nsecond column, we only needed
Dialogue: 0,0:11:39.31,0:11:40.47,Default,,0000,0000,0000,,those three cofactors.
Dialogue: 0,0:11:41.04,0:11:46.47,Default,,0000,0000,0000,,So before we go ahead and work\Nout all nine, if all we want to
Dialogue: 0,0:11:46.47,0:11:51.18,Default,,0000,0000,0000,,do is workout the determinant of\Na, we should decide which row or
Dialogue: 0,0:11:51.18,0:11:55.16,Default,,0000,0000,0000,,column we're going to use first,\Nand then only workout the
Dialogue: 0,0:11:55.16,0:11:59.14,Default,,0000,0000,0000,,cofactors that correspond to the\Nrow or the column that we've
Dialogue: 0,0:11:59.14,0:12:03.96,Default,,0000,0000,0000,,chosen. And in general it's\Nhandy idea. Good idea to keep
Dialogue: 0,0:12:03.96,0:12:08.85,Default,,0000,0000,0000,,your workload down is if you've\Ngot zeros in a row or column.
Dialogue: 0,0:12:08.85,0:12:13.36,Default,,0000,0000,0000,,That's a good row or column to\Nchoose, because you don't need
Dialogue: 0,0:12:13.36,0:12:17.50,Default,,0000,0000,0000,,to actually workout the cofactor\Nthat goes with the zero element,
Dialogue: 0,0:12:17.50,0:12:22.01,Default,,0000,0000,0000,,because it's going to get end up\Nbeing multiplied by that zero
Dialogue: 0,0:12:22.01,0:12:24.64,Default,,0000,0000,0000,,element. So if we use the first
Dialogue: 0,0:12:24.64,0:12:30.09,Default,,0000,0000,0000,,column. The first value is 7,\Nthe next value is zero and the
Dialogue: 0,0:12:30.09,0:12:31.86,Default,,0000,0000,0000,,last value is minus 3.
Dialogue: 0,0:12:32.88,0:12:38.32,Default,,0000,0000,0000,,We then have to multiply these\Nelements by their corresponding
Dialogue: 0,0:12:38.32,0:12:41.04,Default,,0000,0000,0000,,cofactors 7. * -- 2.
Dialogue: 0,0:12:41.57,0:12:44.99,Default,,0000,0000,0000,,Nought Times 8 and 3 * -- 5.
Dialogue: 0,0:12:48.41,0:12:55.39,Default,,0000,0000,0000,,Have them up. We get 7 * -- 2\Nis minus 14 nought. Times 8 is
Dialogue: 0,0:12:55.39,0:12:58.87,Default,,0000,0000,0000,,nought minus 3 * -- 5 is plus
Dialogue: 0,0:12:58.87,0:13:04.30,Default,,0000,0000,0000,,15. An minus 14 plus not plus 15\Nis equal to 1. The same answers
Dialogue: 0,0:13:04.30,0:13:08.75,Default,,0000,0000,0000,,before we knew it was going to\Ncome out to be the same answers
Dialogue: 0,0:13:08.75,0:13:12.25,Default,,0000,0000,0000,,before. The important thing is\Nthat we really didn't need to
Dialogue: 0,0:13:12.25,0:13:16.70,Default,,0000,0000,0000,,know what the code factor of 0\Nwas. So once we've chosen to use
Dialogue: 0,0:13:16.70,0:13:20.52,Default,,0000,0000,0000,,this column, we actually need to\Nfind the cofactors of seven and
Dialogue: 0,0:13:20.52,0:13:24.34,Default,,0000,0000,0000,,minus three because the Co\Nfactor of nought is going to get
Dialogue: 0,0:13:24.34,0:13:28.39,Default,,0000,0000,0000,,multiplied by normal. So what\Nwe've seen then is how to
Dialogue: 0,0:13:28.39,0:13:31.74,Default,,0000,0000,0000,,workout the determinant of a\Nthree by three matrix. What we
Dialogue: 0,0:13:31.74,0:13:35.69,Default,,0000,0000,0000,,do is we pick Aurora column and\Nkeep our work as low as
Dialogue: 0,0:13:35.69,0:13:39.34,Default,,0000,0000,0000,,possible. We choose the row or\Ncolumn with the most zeros, and
Dialogue: 0,0:13:39.34,0:13:42.98,Default,,0000,0000,0000,,then what we do is we workout\Nthe cofactors of every element
Dialogue: 0,0:13:42.98,0:13:46.94,Default,,0000,0000,0000,,in the row or column that we've\Nchosen, but we don't have to
Dialogue: 0,0:13:46.94,0:13:48.46,Default,,0000,0000,0000,,workout the cofactors of zero
Dialogue: 0,0:13:48.46,0:13:52.69,Default,,0000,0000,0000,,elements. Because what we do\Nthen is we multiply each
Dialogue: 0,0:13:52.69,0:13:56.90,Default,,0000,0000,0000,,element by its cofactor, and\Nthen we add up the products of
Dialogue: 0,0:13:56.90,0:14:00.41,Default,,0000,0000,0000,,element times, code factor, and\Nthat's how we workout the
Dialogue: 0,0:14:00.41,0:14:03.57,Default,,0000,0000,0000,,determinant. Now this process\Nworks for any size square
Dialogue: 0,0:14:03.57,0:14:07.08,Default,,0000,0000,0000,,matrix, although once you get\Nabove 3, the amount of
Dialogue: 0,0:14:07.08,0:14:09.54,Default,,0000,0000,0000,,arithmetic involve gets to be\Nvery large.