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Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:05.50,0:00:09.83,Default,,0000,0000,0000,,We've already seen how to use\Nthe inverse of two by two
Dialogue: 0,0:00:09.83,0:00:13.08,Default,,0000,0000,0000,,matrices to solve systems of two\Nsimultaneous equations. Inverse
Dialogue: 0,0:00:13.08,0:00:15.97,Default,,0000,0000,0000,,matrices are always useful in\Nsolving simultaneous equations,
Dialogue: 0,0:00:15.97,0:00:21.38,Default,,0000,0000,0000,,and so we want to look at in\Nthis video is how to find the
Dialogue: 0,0:00:21.38,0:00:25.36,Default,,0000,0000,0000,,inverse of a three by three\Nmatrix. This video builds very
Dialogue: 0,0:00:25.36,0:00:28.96,Default,,0000,0000,0000,,much on the previous video which\Ndescribe finding the determinant
Dialogue: 0,0:00:28.96,0:00:32.94,Default,,0000,0000,0000,,of a three by three matrix, and\Nit's important that you're
Dialogue: 0,0:00:32.94,0:00:35.10,Default,,0000,0000,0000,,familiar with the ideas in that
Dialogue: 0,0:00:35.10,0:00:40.25,Default,,0000,0000,0000,,video. Before you watch this one\Nin particular, you need to know
Dialogue: 0,0:00:40.25,0:00:41.54,Default,,0000,0000,0000,,about cofactors and
Dialogue: 0,0:00:41.54,0:00:46.33,Default,,0000,0000,0000,,determinants. Here's the Matrix\Na that we saw in the video on
Dialogue: 0,0:00:46.33,0:00:49.79,Default,,0000,0000,0000,,calculating determinants and we\Nsaw in that video how every
Dialogue: 0,0:00:49.79,0:00:52.21,Default,,0000,0000,0000,,element in the matrix A has its
Dialogue: 0,0:00:52.21,0:00:56.35,Default,,0000,0000,0000,,own cofactor. And the Co\Nfactor is just a value, a
Dialogue: 0,0:00:56.35,0:00:59.49,Default,,0000,0000,0000,,single number, and what I've\Ndone here is I've assembled
Dialogue: 0,0:00:59.49,0:01:02.63,Default,,0000,0000,0000,,all those cofactors into a\Nmatrix that we've called see.
Dialogue: 0,0:01:02.63,0:01:06.08,Default,,0000,0000,0000,,So for instance, the cofactor\Nof elements 7 is minus two,
Dialogue: 0,0:01:06.08,0:01:09.85,Default,,0000,0000,0000,,and the cofactor Element 4 is\N7, and so it goes on.
Dialogue: 0,0:01:11.86,0:01:17.21,Default,,0000,0000,0000,,Now what we want to find the\Ninverse of matrix A. We have to
Dialogue: 0,0:01:17.21,0:01:22.94,Default,,0000,0000,0000,,use this matrix C, but not quite\Nhow it is at the moment. What we
Dialogue: 0,0:01:22.94,0:01:27.52,Default,,0000,0000,0000,,have to do is we use it to\Ncreate something called the
Dialogue: 0,0:01:27.52,0:01:32.11,Default,,0000,0000,0000,,adjoint matrix and we call the\Nadjoint matrix adj adj for a
Dialogue: 0,0:01:32.11,0:01:36.69,Default,,0000,0000,0000,,joint. The adjoint of matrix A.\NThis is true for all matrices,
Dialogue: 0,0:01:36.69,0:01:40.51,Default,,0000,0000,0000,,not just this matrix is the\Ntranspose of the cofactor
Dialogue: 0,0:01:40.51,0:01:42.42,Default,,0000,0000,0000,,matrix. So here's the cofactor
Dialogue: 0,0:01:42.42,0:01:46.84,Default,,0000,0000,0000,,matrix. To find the adjoint\Nmatrix we have to transpose
Dialogue: 0,0:01:46.84,0:01:51.64,Default,,0000,0000,0000,,this. That means we have to\Nchange the rows into columns.
Dialogue: 0,0:01:51.64,0:01:56.87,Default,,0000,0000,0000,,The columns into rows. So the\Nfirst row minus 239 becomes the
Dialogue: 0,0:01:56.87,0:02:01.67,Default,,0000,0000,0000,,first column minus 239. The 2nd\Nrow becomes the second column.
Dialogue: 0,0:02:02.25,0:02:08.37,Default,,0000,0000,0000,,And the third row becomes\Nthe third column.
Dialogue: 0,0:02:09.05,0:02:16.38,Default,,0000,0000,0000,,So we found now the adjoint\Nmatrix, then the formula for the
Dialogue: 0,0:02:16.38,0:02:23.52,Default,,0000,0000,0000,,inverse matrix. Is the inverse\Nof matrix A? Is one over
Dialogue: 0,0:02:23.52,0:02:25.89,Default,,0000,0000,0000,,the determinant of a?
Dialogue: 0,0:02:26.64,0:02:29.92,Default,,0000,0000,0000,,Times by the adjoint matrix.
Dialogue: 0,0:02:31.10,0:02:37.88,Default,,0000,0000,0000,,I'm writing that using\Nthis notation. A inverse
Dialogue: 0,0:02:37.88,0:02:44.67,Default,,0000,0000,0000,,is the value one\Nover determinant of a
Dialogue: 0,0:02:44.67,0:02:50.60,Default,,0000,0000,0000,,times by the adjoint\Nmatrix of a.
Dialogue: 0,0:02:51.79,0:02:55.35,Default,,0000,0000,0000,,And that's the key result
Dialogue: 0,0:02:55.35,0:03:01.61,Default,,0000,0000,0000,,in finding. The inverse of any\Nmatrix and will use that result
Dialogue: 0,0:03:01.61,0:03:04.51,Default,,0000,0000,0000,,to find the inverse of our
Dialogue: 0,0:03:04.51,0:03:09.73,Default,,0000,0000,0000,,matrix A. Here's our Matrix A\Nand we've worked out the adjoint
Dialogue: 0,0:03:09.73,0:03:14.12,Default,,0000,0000,0000,,matrix. The formula for the\Ninverse is the inverse of A is
Dialogue: 0,0:03:14.12,0:03:18.51,Default,,0000,0000,0000,,one over the determinant value\Ntimes by the adjoint of a. Now
Dialogue: 0,0:03:18.51,0:03:22.91,Default,,0000,0000,0000,,in the video where we worked out\Nthe determinant of this matrix,
Dialogue: 0,0:03:22.91,0:03:27.30,Default,,0000,0000,0000,,we found the determinant of this\Nmatrix A is equal to 1.
Dialogue: 0,0:03:28.11,0:03:33.83,Default,,0000,0000,0000,,So in this case, the value 1\Ndivided by the determinant of a
Dialogue: 0,0:03:33.83,0:03:40.87,Default,,0000,0000,0000,,is just 1 / 1 which is 1. So\Nin this case a inverse is 1
Dialogue: 0,0:03:40.87,0:03:47.47,Default,,0000,0000,0000,,times the joint of a or just the\Nadjoint of a. So a inverse turns
Dialogue: 0,0:03:47.47,0:03:53.63,Default,,0000,0000,0000,,out to be just this matrix here\Nminus two 8 -- 5, three minus
Dialogue: 0,0:03:53.63,0:03:55.83,Default,,0000,0000,0000,,11, seven 9 -- 3421.
Dialogue: 0,0:03:56.39,0:04:00.67,Default,,0000,0000,0000,,What you should do is you\Nshould check by doing matrix
Dialogue: 0,0:04:00.67,0:04:04.17,Default,,0000,0000,0000,,multiplication that when you\Nmultiply original matrix A by
Dialogue: 0,0:04:04.17,0:04:08.84,Default,,0000,0000,0000,,this matrix we've just found\Nhere a inverse, so we do a
Dialogue: 0,0:04:08.84,0:04:13.12,Default,,0000,0000,0000,,multiplied by inverse that you\Ndo indeed get the three by
Dialogue: 0,0:04:13.12,0:04:14.28,Default,,0000,0000,0000,,three identity matrix.
Dialogue: 0,0:04:15.38,0:04:20.80,Default,,0000,0000,0000,,What I want to go on to do now\Nis to show how we can use this
Dialogue: 0,0:04:20.80,0:04:23.04,Default,,0000,0000,0000,,inverse matrix to solve a set of
Dialogue: 0,0:04:23.04,0:04:27.46,Default,,0000,0000,0000,,simultaneous linear equations.\NHere we have a set of three
Dialogue: 0,0:04:27.46,0:04:32.71,Default,,0000,0000,0000,,simultaneous equations, 7X plus\Ntwo Y + Z = 21, three y --
Dialogue: 0,0:04:32.71,0:04:39.98,Default,,0000,0000,0000,,8 = 5 minus three X + 4 Y\Nminus two, Z = -- 1. We want to
Dialogue: 0,0:04:39.98,0:04:44.02,Default,,0000,0000,0000,,solve these define the values of\NXY&Z. There are unknowns.
Dialogue: 0,0:04:44.81,0:04:49.31,Default,,0000,0000,0000,,And we've seen how we can do. We\Ncan represent these equations
Dialogue: 0,0:04:49.31,0:04:53.81,Default,,0000,0000,0000,,using some key matrices, so we\Nhave matrix A, which is the
Dialogue: 0,0:04:53.81,0:04:58.31,Default,,0000,0000,0000,,matrix of coefficients 721\Nnought. 3 -- 1 -- 3, four and
Dialogue: 0,0:04:58.31,0:05:01.60,Default,,0000,0000,0000,,minus 2. There's the question\Nhere is not because we
Dialogue: 0,0:05:01.60,0:05:04.11,Default,,0000,0000,0000,,haven't got any access, so\Nwe've got like no XSS.
Dialogue: 0,0:05:05.20,0:05:10.19,Default,,0000,0000,0000,,Then we have a matrix which is\Njust a single column, which is
Dialogue: 0,0:05:10.19,0:05:14.80,Default,,0000,0000,0000,,the unknowns XY&Z and then I\Nhave a separate matrix B which
Dialogue: 0,0:05:14.80,0:05:19.41,Default,,0000,0000,0000,,again is just a single column\Nwith the values from the right
Dialogue: 0,0:05:19.41,0:05:23.63,Default,,0000,0000,0000,,hand side of the equations. So\Nin matrix form these equations
Dialogue: 0,0:05:23.63,0:05:29.01,Default,,0000,0000,0000,,can be written as this matrix a\Ntimes this vector X is equal to
Dialogue: 0,0:05:29.01,0:05:30.16,Default,,0000,0000,0000,,this matrix disks.
Dialogue: 0,0:05:31.83,0:05:35.52,Default,,0000,0000,0000,,Then the solution of this\Nequation ax equals B.
Dialogue: 0,0:05:36.26,0:05:41.17,Default,,0000,0000,0000,,We can find by multiplying\Nboth sides by the inverse of A
Dialogue: 0,0:05:41.17,0:05:46.48,Default,,0000,0000,0000,,to get that X = a inverse\Ntimes D. So to find our
Dialogue: 0,0:05:46.48,0:05:51.80,Default,,0000,0000,0000,,solution XY and Z, we need to\Nfind a inverse. And of course
Dialogue: 0,0:05:51.80,0:05:56.30,Default,,0000,0000,0000,,we've just done that. We've\Nseen that a inverse is this
Dialogue: 0,0:05:56.30,0:05:56.71,Default,,0000,0000,0000,,matrix.
Dialogue: 0,0:05:57.95,0:05:59.21,Default,,0000,0000,0000,,Minus 2.
Dialogue: 0,0:06:00.28,0:06:06.36,Default,,0000,0000,0000,,8 -- 5 three\Nminus 11 seven.
Dialogue: 0,0:06:06.90,0:06:10.23,Default,,0000,0000,0000,,9 --
Dialogue: 0,0:06:10.23,0:06:15.12,Default,,0000,0000,0000,,3421. So that's the matrix, a\Ninverse that we've just seen
Dialogue: 0,0:06:15.12,0:06:16.40,Default,,0000,0000,0000,,a few minutes ago.
Dialogue: 0,0:06:18.42,0:06:23.31,Default,,0000,0000,0000,,And so we have to do a inverse\Ntimes be. So here's be.
Dialogue: 0,0:06:23.92,0:06:27.58,Default,,0000,0000,0000,,20 one 5 --
Dialogue: 0,0:06:27.58,0:06:34.41,Default,,0000,0000,0000,,1. And so we do the matrix\Nmultiplication 2 * 21 is minus
Dialogue: 0,0:06:34.41,0:06:41.31,Default,,0000,0000,0000,,40, two 8 * 5 is 40 --\N5 * -- 1 is +5.
Dialogue: 0,0:06:42.04,0:06:43.55,Default,,0000,0000,0000,,That's our first entry.
Dialogue: 0,0:06:44.07,0:06:51.57,Default,,0000,0000,0000,,I'm going along the 2nd row 3 *\N2163 -- 11 * 5 is minus 50 five
Dialogue: 0,0:06:51.57,0:06:54.65,Default,,0000,0000,0000,,7 * -- 1 is minus 7.
Dialogue: 0,0:06:55.17,0:07:02.80,Default,,0000,0000,0000,,And along the last row, 9 *\N21 is 189 -- 34 * 5
Dialogue: 0,0:07:02.80,0:07:08.80,Default,,0000,0000,0000,,is minus 170 and 21 * --\N1 is minus 21.
Dialogue: 0,0:07:09.32,0:07:16.01,Default,,0000,0000,0000,,And so if we do the arithmetic,\Nwe have 45 -- 42, which is 3.
Dialogue: 0,0:07:16.01,0:07:19.58,Default,,0000,0000,0000,,We have 63 -- 6 D 2 which
Dialogue: 0,0:07:19.58,0:07:26.68,Default,,0000,0000,0000,,is 1. And we have 189\N-- 191, which is minus 2. So
Dialogue: 0,0:07:26.68,0:07:32.24,Default,,0000,0000,0000,,the unknowns that we're trying\Nto find the column matrix X.
Dialogue: 0,0:07:32.77,0:07:36.84,Default,,0000,0000,0000,,Which is XY\Nzed?
Dialogue: 0,0:07:38.02,0:07:44.78,Default,,0000,0000,0000,,Is this column matrix three 1 --\N2 so X is equal to 3, Y
Dialogue: 0,0:07:44.78,0:07:51.10,Default,,0000,0000,0000,,is equal to 1 that is equal to\Nminus two. That's our solution X
Dialogue: 0,0:07:51.10,0:07:54.26,Default,,0000,0000,0000,,= 3 Y equals one, zed equals
Dialogue: 0,0:07:54.26,0:07:58.87,Default,,0000,0000,0000,,minus 2. And you can check if\Nyou substitute these values
Dialogue: 0,0:07:58.87,0:08:01.100,Default,,0000,0000,0000,,back into any of these\Nequations, you'll see that the
Dialogue: 0,0:08:01.100,0:08:05.13,Default,,0000,0000,0000,,two sides do balance. So for\Ninstance, just the second
Dialogue: 0,0:08:05.13,0:08:09.20,Default,,0000,0000,0000,,equation with. Why is worn and\Nsaid he's minus two, we get 3
Dialogue: 0,0:08:09.20,0:08:13.89,Default,,0000,0000,0000,,* 1 is 3 -- -- 2 three plus\Ntwo, which is indeed five you
Dialogue: 0,0:08:13.89,0:08:17.33,Default,,0000,0000,0000,,submit those into the two.\NYou'll see that they work as
Dialogue: 0,0:08:17.33,0:08:17.65,Default,,0000,0000,0000,,well.
Dialogue: 0,0:08:18.84,0:08:23.35,Default,,0000,0000,0000,,So inverse matrices are really\Nimportant when it comes to
Dialogue: 0,0:08:23.35,0:08:26.96,Default,,0000,0000,0000,,solving simultaneous equations.\NNow you'll notice that our
Dialogue: 0,0:08:26.96,0:08:31.92,Default,,0000,0000,0000,,formula for finding the inverse\Nfor three by three matrix had
Dialogue: 0,0:08:31.92,0:08:35.98,Default,,0000,0000,0000,,the value 1 divided by the\Ndeterminant of a.
Dialogue: 0,0:08:36.51,0:08:40.16,Default,,0000,0000,0000,,Now that means that if the\Ndetermined today is zero, we
Dialogue: 0,0:08:40.16,0:08:43.81,Default,,0000,0000,0000,,can't actually work that value\Nout because we can't divide by
Dialogue: 0,0:08:43.81,0:08:48.39,Default,,0000,0000,0000,,zero. Now, when the determined\Nto the matrix is zero, we say
Dialogue: 0,0:08:48.39,0:08:52.20,Default,,0000,0000,0000,,the matrix is singular and when\Na matrix is singular, it doesn't
Dialogue: 0,0:08:52.20,0:08:55.07,Default,,0000,0000,0000,,have an inverse matrix and\Ninverse matrix just doesn't
Dialogue: 0,0:08:55.07,0:08:58.56,Default,,0000,0000,0000,,exist, and so we can't apply the\Nformula. So whenever we're
Dialogue: 0,0:08:58.56,0:09:01.74,Default,,0000,0000,0000,,trying to workout inverse\Nmatrices, the thing we should do
Dialogue: 0,0:09:01.74,0:09:05.24,Default,,0000,0000,0000,,first of all is workout the\Ndeterminant and check that it
Dialogue: 0,0:09:05.24,0:09:07.79,Default,,0000,0000,0000,,isn't going to turn out to be 0.