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Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:05.48,0:00:09.11,Default,,0000,0000,0000,,Once you know how to multiply 2\Nmatrices together, a natural
Dialogue: 0,0:00:09.11,0:00:13.40,Default,,0000,0000,0000,,question to ask is whether or\Nnot one matrix can be divided by
Dialogue: 0,0:00:13.40,0:00:17.36,Default,,0000,0000,0000,,another matrix, and the answer\Nto that question is no, there is
Dialogue: 0,0:00:17.36,0:00:19.01,Default,,0000,0000,0000,,no such thing as matrix
Dialogue: 0,0:00:19.01,0:00:22.08,Default,,0000,0000,0000,,division. Nevertheless, there's\Nanother operation that we can
Dialogue: 0,0:00:22.08,0:00:25.11,Default,,0000,0000,0000,,introduce which plays a very\Nsimilar role to division, and
Dialogue: 0,0:00:25.11,0:00:28.75,Default,,0000,0000,0000,,it's called finding the inverse\Nof a matrix. So in this video
Dialogue: 0,0:00:28.75,0:00:32.69,Default,,0000,0000,0000,,I'm going to explain what we\Nmean by the inverse of a matrix.
Dialogue: 0,0:00:33.83,0:00:37.26,Default,,0000,0000,0000,,To start with, let's just look\Nat these two matrices that I've
Dialogue: 0,0:00:37.26,0:00:40.41,Default,,0000,0000,0000,,written down here, and let's\Nfind the product of them. We
Dialogue: 0,0:00:40.41,0:00:41.55,Default,,0000,0000,0000,,multiply these two together.
Dialogue: 0,0:00:42.59,0:00:49.83,Default,,0000,0000,0000,,4 * 1 is 4 added to 3 *\N-- 1 is 4 -- 3 which is 1.
Dialogue: 0,0:00:51.40,0:00:57.93,Default,,0000,0000,0000,,4 * -- 3 is minus 12, three 4 +\N12 -- 12 + 12 is 0.
Dialogue: 0,0:00:59.76,0:01:05.90,Default,,0000,0000,0000,,1 * 1 is 1 and 1 * -- 1 is minus\None 1 -- 1 is or is not.
Dialogue: 0,0:01:07.01,0:01:12.26,Default,,0000,0000,0000,,1 * -- 3 is minus three 1 * 4 is\Nfour 4 -- 3 is 1.
Dialogue: 0,0:01:13.97,0:01:18.09,Default,,0000,0000,0000,,The point I'm trying to make is\Nthat when we multiply these two
Dialogue: 0,0:01:18.09,0:01:21.26,Default,,0000,0000,0000,,matrices together, the answer\Nthat we get is an identity
Dialogue: 0,0:01:21.26,0:01:22.85,Default,,0000,0000,0000,,matrix. That point is important.
Dialogue: 0,0:01:26.89,0:01:29.88,Default,,0000,0000,0000,,Suppose we do the\Nmultiplication in the opposite
Dialogue: 0,0:01:29.88,0:01:34.74,Default,,0000,0000,0000,,order, so 1 -- 3 -- 1 four\Nmultiplied by 4311. So I've
Dialogue: 0,0:01:34.74,0:01:37.36,Default,,0000,0000,0000,,changed the order of the\Nmultiplication there.
Dialogue: 0,0:01:38.57,0:01:43.25,Default,,0000,0000,0000,,One's voice 4 -- 3 * 1 is minus\Nthree. 4 -- 3 is 1.
Dialogue: 0,0:01:44.27,0:01:49.57,Default,,0000,0000,0000,,Once three is 3 -- 3 * 1 is\Nminus three, 3 -- 3 is nothing.
Dialogue: 0,0:01:50.57,0:01:55.61,Default,,0000,0000,0000,,Minus 1 * 4 is minus 44144 and\Nthe result of adding them
Dialogue: 0,0:01:55.61,0:01:56.78,Default,,0000,0000,0000,,together is 0.
Dialogue: 0,0:01:57.85,0:02:04.07,Default,,0000,0000,0000,,Minus 1 * 3 is minus 3414, four\N4 -- 3 is 1, so again the result
Dialogue: 0,0:02:04.07,0:02:07.37,Default,,0000,0000,0000,,of multiplying these two\Ntogether is an identity matrix.
Dialogue: 0,0:02:08.46,0:02:12.31,Default,,0000,0000,0000,,This leads us to the following\Ndefinition for the inverse of a
Dialogue: 0,0:02:12.31,0:02:16.16,Default,,0000,0000,0000,,matrix. If you've got one matrix\Nand you multiply it by another
Dialogue: 0,0:02:16.16,0:02:19.70,Default,,0000,0000,0000,,matrix, and the answer is an\Nidentity matrix, then the second
Dialogue: 0,0:02:19.70,0:02:23.55,Default,,0000,0000,0000,,matrix is the inverse of the\Nfirst matrix and vice versa. The
Dialogue: 0,0:02:23.55,0:02:27.08,Default,,0000,0000,0000,,first matrix is the inverse of\Nthe second matrix. Now suppose
Dialogue: 0,0:02:27.08,0:02:31.25,Default,,0000,0000,0000,,we have a matrix A and we\Nmultiply it by its inverse and
Dialogue: 0,0:02:31.25,0:02:35.42,Default,,0000,0000,0000,,we'll denote the inverse by A to\Nthe power minus one. Now that's
Dialogue: 0,0:02:35.42,0:02:38.96,Default,,0000,0000,0000,,not really a power. That does\Nnot mean 1 / A.
Dialogue: 0,0:02:39.03,0:02:42.33,Default,,0000,0000,0000,,This is the notation that\Nwill use for the inverse of
Dialogue: 0,0:02:42.33,0:02:46.23,Default,,0000,0000,0000,,the matrix A, so if you have\Na matrix a multiplied by its
Dialogue: 0,0:02:46.23,0:02:46.53,Default,,0000,0000,0000,,inverse.
Dialogue: 0,0:02:48.71,0:02:52.19,Default,,0000,0000,0000,,The answer is an identity\Nmatrix. That's what we mean
Dialogue: 0,0:02:52.19,0:02:55.67,Default,,0000,0000,0000,,by an inverse matrix. The\Nsame works the other way
Dialogue: 0,0:02:55.67,0:02:59.15,Default,,0000,0000,0000,,around as well, so if you\Nstarted with an inverse
Dialogue: 0,0:02:59.15,0:03:02.28,Default,,0000,0000,0000,,matrix and multiplied it by\Nthe original matrix, the
Dialogue: 0,0:03:02.28,0:03:05.41,Default,,0000,0000,0000,,answer two would be an\Nidentity matrix, and that's
Dialogue: 0,0:03:05.41,0:03:07.85,Default,,0000,0000,0000,,the important definition of\Nan identity matrix.
Dialogue: 0,0:03:09.23,0:03:13.16,Default,,0000,0000,0000,,If we want to find an inverse of\Nthe matrix, there's a simple
Dialogue: 0,0:03:13.16,0:03:16.78,Default,,0000,0000,0000,,formula that exists when the\Nmatrix is a two by two matrix.
Dialogue: 0,0:03:16.78,0:03:18.89,Default,,0000,0000,0000,,So let's have a look at the
Dialogue: 0,0:03:18.89,0:03:25.53,Default,,0000,0000,0000,,formula Now. Suppose we've\Ngot the matrix ABCD.
Dialogue: 0,0:03:26.70,0:03:29.70,Default,,0000,0000,0000,,The formula for the inverse\Nis as follows. The inverse,
Dialogue: 0,0:03:29.70,0:03:33.30,Default,,0000,0000,0000,,which as I say is denoted by\NA to the minus one.
Dialogue: 0,0:03:35.83,0:03:37.97,Default,,0000,0000,0000,,Is given by 1 divided by.
Dialogue: 0,0:03:38.61,0:03:43.65,Default,,0000,0000,0000,,The number is 8 * D. Subtract B\N* C rather strange formula, but
Dialogue: 0,0:03:43.65,0:03:48.69,Default,,0000,0000,0000,,you'll see how it develops in a\Nminute. 1 / a D minus BC
Dialogue: 0,0:03:48.69,0:03:52.65,Default,,0000,0000,0000,,multiplied by the matrix that\Nyou get when you interchange the
Dialogue: 0,0:03:52.65,0:03:58.05,Default,,0000,0000,0000,,A and the D round. So the DB\Ncomes up here and the A goes
Dialogue: 0,0:03:58.05,0:04:02.79,Default,,0000,0000,0000,,down there. And we changed the\Nsign but leave leave these
Dialogue: 0,0:04:02.79,0:04:07.10,Default,,0000,0000,0000,,elements in place. The BNC in\Nplace, but change their sign to
Dialogue: 0,0:04:07.10,0:04:11.77,Default,,0000,0000,0000,,minus BN minus C, and this\Nformula that we have here is the
Dialogue: 0,0:04:11.77,0:04:14.100,Default,,0000,0000,0000,,formula. Then for the inverse of\Nthe matrix A.
Dialogue: 0,0:04:16.63,0:04:20.85,Default,,0000,0000,0000,,I'm going to apply this\Nformula to try and find the
Dialogue: 0,0:04:20.85,0:04:25.08,Default,,0000,0000,0000,,inverse of a matrix that I'll\Ngive you here. Now, supposing
Dialogue: 0,0:04:25.08,0:04:27.38,Default,,0000,0000,0000,,the Matrix is a is 3142.
Dialogue: 0,0:04:30.24,0:04:34.14,Default,,0000,0000,0000,,Now applying the formula, we'll\Nget that the inverse matrix is
Dialogue: 0,0:04:34.14,0:04:41.24,Default,,0000,0000,0000,,equal to 1 / 8 * D. That's 3 *\N2, which is 6 -- B * C, which is
Dialogue: 0,0:04:41.24,0:04:42.31,Default,,0000,0000,0000,,1 * 4.
Dialogue: 0,0:04:44.01,0:04:47.37,Default,,0000,0000,0000,,Multiplied by a matrix which you\Nget when you interchange the A
Dialogue: 0,0:04:47.37,0:04:49.89,Default,,0000,0000,0000,,and the D. So the A and the D
Dialogue: 0,0:04:49.89,0:04:53.64,Default,,0000,0000,0000,,swap round. So swapping the\Nthree in the two round will
Dialogue: 0,0:04:53.64,0:04:54.69,Default,,0000,0000,0000,,get two and three.
Dialogue: 0,0:04:56.01,0:04:58.72,Default,,0000,0000,0000,,And changing the sign of the\Nother two elements, but leaving
Dialogue: 0,0:04:58.72,0:04:59.95,Default,,0000,0000,0000,,them in the same place.
Dialogue: 0,0:05:02.44,0:05:04.48,Default,,0000,0000,0000,,So this is the inverse of matrix
Dialogue: 0,0:05:04.48,0:05:08.94,Default,,0000,0000,0000,,A. Let's just tidy it\Nup a little bit 1 / 6
Dialogue: 0,0:05:08.94,0:05:11.57,Default,,0000,0000,0000,,-- 4 is 1 / 2 or 1/2.
Dialogue: 0,0:05:15.71,0:05:18.83,Default,,0000,0000,0000,,And we could leave the answer\Nlike that. Or we can multiply
Dialogue: 0,0:05:18.83,0:05:21.17,Default,,0000,0000,0000,,the factor of 1/2 inside,\Nmultiplying every element inside
Dialogue: 0,0:05:21.17,0:05:24.55,Default,,0000,0000,0000,,by the half. But I'm going to\Nleave it like that for now.
Dialogue: 0,0:05:25.28,0:05:29.05,Default,,0000,0000,0000,,So this is the inverse of matrix\NA. If we want to check whether
Dialogue: 0,0:05:29.05,0:05:32.54,Default,,0000,0000,0000,,it's whether it's the right\Nanswer or not, all we have to do
Dialogue: 0,0:05:32.54,0:05:35.50,Default,,0000,0000,0000,,is multiply these two together\Nand the answer that we should
Dialogue: 0,0:05:35.50,0:05:38.46,Default,,0000,0000,0000,,get should be an identity\Nmatrix. So let's just check that
Dialogue: 0,0:05:38.46,0:05:41.96,Default,,0000,0000,0000,,what I'll do is I'll workout A\Nto the minus 1 * A.
Dialogue: 0,0:05:43.88,0:05:46.83,Default,,0000,0000,0000,,So here's A to the minus\None we've just found.
Dialogue: 0,0:05:49.25,0:05:53.47,Default,,0000,0000,0000,,And I'm going to multiply it by\Na, which was three 142.
Dialogue: 0,0:05:55.97,0:05:58.14,Default,,0000,0000,0000,,Let's leave the half\Noutside for now.
Dialogue: 0,0:05:59.68,0:06:03.82,Default,,0000,0000,0000,,To do the matrix multiplication,\Nhere we've got 2, three or six.
Dialogue: 0,0:06:04.79,0:06:09.29,Default,,0000,0000,0000,,Added 2 -- 1 * 4 that's 6 -- 4\Nwhich is 2.
Dialogue: 0,0:06:10.66,0:06:11.76,Default,,0000,0000,0000,,2 ones or two.
Dialogue: 0,0:06:12.42,0:06:16.30,Default,,0000,0000,0000,,Add it to minus 1 * 2. That's 2\N-- 2, which is nothing.
Dialogue: 0,0:06:17.67,0:06:19.28,Default,,0000,0000,0000,,Minus 4 * 3.
Dialogue: 0,0:06:19.94,0:06:25.43,Default,,0000,0000,0000,,Plus 3 * 4, which is minus 12 +\N12, which is nothing and finally
Dialogue: 0,0:06:25.43,0:06:32.02,Default,,0000,0000,0000,,minus 4 * 1, which is minus four\Nadded to 3, two six 6 -- 4 is 2.
Dialogue: 0,0:06:33.43,0:06:37.52,Default,,0000,0000,0000,,And finally, if we multiply each\Nelement by the factor of 1/2
Dialogue: 0,0:06:37.52,0:06:41.61,Default,,0000,0000,0000,,outside, you'll see that we get\Nthe identity matrix, so that's a
Dialogue: 0,0:06:41.61,0:06:45.36,Default,,0000,0000,0000,,verification that this matrix we\Nhave found here is the inverse
Dialogue: 0,0:06:45.36,0:06:47.41,Default,,0000,0000,0000,,of the matrix we started with.
Dialogue: 0,0:06:48.13,0:06:51.39,Default,,0000,0000,0000,,For yourself, you could verify\Nthat we get the same results by
Dialogue: 0,0:06:51.39,0:06:54.11,Default,,0000,0000,0000,,multiplying them the opposite\Nway around. Finding a time Zeta
Dialogue: 0,0:06:54.11,0:06:57.38,Default,,0000,0000,0000,,minus one, and you'll still find\Nthat you get an identity matrix,
Dialogue: 0,0:06:57.38,0:07:00.91,Default,,0000,0000,0000,,and that's how you use the\Nformula to find the inverse of a
Dialogue: 0,0:07:00.91,0:07:02.00,Default,,0000,0000,0000,,two by two matrix.
Dialogue: 0,0:07:03.22,0:07:06.94,Default,,0000,0000,0000,,Now there are certain situations\Nwhere this won't work, and the
Dialogue: 0,0:07:06.94,0:07:10.66,Default,,0000,0000,0000,,reason why it won't work is\Nobtained by looking at this
Dialogue: 0,0:07:10.66,0:07:12.35,Default,,0000,0000,0000,,quantity in the formula here.
Dialogue: 0,0:07:13.33,0:07:16.89,Default,,0000,0000,0000,,If it transpires that a D -- B C
Dialogue: 0,0:07:16.89,0:07:21.30,Default,,0000,0000,0000,,is 0. Would be trying to divide\Nby zero here.
Dialogue: 0,0:07:21.96,0:07:23.80,Default,,0000,0000,0000,,And division by zero is never
Dialogue: 0,0:07:23.80,0:07:30.56,Default,,0000,0000,0000,,possible. So what this means is,\Nif ABC&D are such that a D -- B
Dialogue: 0,0:07:30.56,0:07:31.79,Default,,0000,0000,0000,,C is 0.
Dialogue: 0,0:07:32.47,0:07:34.87,Default,,0000,0000,0000,,The inverse of this matrix\Nwill not exist.
Dialogue: 0,0:07:35.92,0:07:39.84,Default,,0000,0000,0000,,Now you may remember that the\Nquantity AD minus BC we've met
Dialogue: 0,0:07:39.84,0:07:42.79,Default,,0000,0000,0000,,before. It's called the\Ndeterminant of this matrix A.
Dialogue: 0,0:07:43.98,0:07:47.34,Default,,0000,0000,0000,,So whenever the determinant of\Nthe matrix A is 0.
Dialogue: 0,0:07:48.24,0:07:49.88,Default,,0000,0000,0000,,The inverse won't exist.
Dialogue: 0,0:07:50.85,0:07:55.15,Default,,0000,0000,0000,,Equivalently, whenever A is a\Nsingular matrix, the inverse
Dialogue: 0,0:07:55.15,0:08:01.37,Default,,0000,0000,0000,,won't exist. Let's just have a\Nquick look at an example of that
Dialogue: 0,0:08:01.37,0:08:06.62,Default,,0000,0000,0000,,sort, supposing we had a matrix\NA which was 326 fourth.
Dialogue: 0,0:08:08.33,0:08:13.90,Default,,0000,0000,0000,,If we find a D minus BC for this\Nmatrix, that's 3 * 4, which is
Dialogue: 0,0:08:13.90,0:08:19.47,Default,,0000,0000,0000,,12. Subtract 2 * 6 which is 1212\N-- 12 is zero. We see that the
Dialogue: 0,0:08:19.47,0:08:21.21,Default,,0000,0000,0000,,determinant of a is 0.
Dialogue: 0,0:08:22.76,0:08:26.95,Default,,0000,0000,0000,,This is a singular matrix, and\Nso because it's a singular
Dialogue: 0,0:08:26.95,0:08:28.86,Default,,0000,0000,0000,,matrix, no inverse matrix will
Dialogue: 0,0:08:28.86,0:08:34.07,Default,,0000,0000,0000,,exist. So the point to remember\Nis that not all two by two
Dialogue: 0,0:08:34.07,0:08:37.89,Default,,0000,0000,0000,,matrices will have an inverse.\NThose that don't have an inverse
Dialogue: 0,0:08:37.89,0:08:41.37,Default,,0000,0000,0000,,accord singular matrices. But\Nwhen a matrix does have an
Dialogue: 0,0:08:41.37,0:08:44.51,Default,,0000,0000,0000,,inverse, this formula will\Nenable you to find it.