[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:05.14,0:00:08.21,Default,,0000,0000,0000,,In this second of the videos on\Nmatrix multiplication, we're
Dialogue: 0,0:00:08.21,0:00:11.28,Default,,0000,0000,0000,,going to delve a little bit more\Ndeeply into matrix
Dialogue: 0,0:00:11.28,0:00:14.35,Default,,0000,0000,0000,,multiplication and look at some\Nof the properties and the
Dialogue: 0,0:00:14.35,0:00:17.11,Default,,0000,0000,0000,,conditions under which different\Nsorts of multiplication can be
Dialogue: 0,0:00:17.11,0:00:20.49,Default,,0000,0000,0000,,carried out. Let's start by\Nlooking at looking at a specific
Dialogue: 0,0:00:20.49,0:00:23.56,Default,,0000,0000,0000,,example in this example. Here\NI've written down to matrices
Dialogue: 0,0:00:23.56,0:00:28.79,Default,,0000,0000,0000,,M&N. And let's look at the sizes\Nof these matrices. The first
Dialogue: 0,0:00:28.79,0:00:33.39,Default,,0000,0000,0000,,matrix M. Is a three row\Nthree column matrix, so
Dialogue: 0,0:00:33.39,0:00:34.85,Default,,0000,0000,0000,,it's three by three.
Dialogue: 0,0:00:37.71,0:00:42.07,Default,,0000,0000,0000,,And the second matrix N is 3\Nrows, two columns.
Dialogue: 0,0:00:42.90,0:00:44.25,Default,,0000,0000,0000,,So it's a 3 by 2.
Dialogue: 0,0:00:46.92,0:00:49.15,Default,,0000,0000,0000,,And we notice that these\Nnumbers are the same.
Dialogue: 0,0:00:50.92,0:00:54.26,Default,,0000,0000,0000,,The number of columns in the\Nfirst is the same as the number
Dialogue: 0,0:00:54.26,0:00:57.09,Default,,0000,0000,0000,,of rows in the second, so we can\Nperform this matrix
Dialogue: 0,0:00:57.09,0:01:00.17,Default,,0000,0000,0000,,multiplication and the size of\Nthe answer will be a three by
Dialogue: 0,0:01:00.17,0:01:03.51,Default,,0000,0000,0000,,two matrix. So right at the\Nstart we know the size of the
Dialogue: 0,0:01:03.51,0:01:06.60,Default,,0000,0000,0000,,answer. It's going to have three\Nrows and two columns just like
Dialogue: 0,0:01:06.60,0:01:09.79,Default,,0000,0000,0000,,this one had. So the shape\Nof the answer is.
Dialogue: 0,0:01:10.91,0:01:13.40,Default,,0000,0000,0000,,Like we have here and we're\Nlooking for these 6 numbers.
Dialogue: 0,0:01:13.98,0:01:14.76,Default,,0000,0000,0000,,In the product.
Dialogue: 0,0:01:16.51,0:01:17.82,Default,,0000,0000,0000,,Let's try and work it out.
Dialogue: 0,0:01:18.78,0:01:23.24,Default,,0000,0000,0000,,To find the number that's in the\Nfirst row, first column. We work
Dialogue: 0,0:01:23.24,0:01:27.70,Default,,0000,0000,0000,,with the first row of the first\Nmatrix and the first column of
Dialogue: 0,0:01:27.70,0:01:28.73,Default,,0000,0000,0000,,the Second Matrix.
Dialogue: 0,0:01:29.51,0:01:33.57,Default,,0000,0000,0000,,What we want is 3\N* 1 which is 3.
Dialogue: 0,0:01:34.58,0:01:41.12,Default,,0000,0000,0000,,2 * -- 2 which is minus 4 and 1\N* 3, which is 3. So we've got 3
Dialogue: 0,0:01:41.12,0:01:42.15,Default,,0000,0000,0000,,ones or three.
Dialogue: 0,0:01:43.73,0:01:50.32,Default,,0000,0000,0000,,2 * -- 2 is minus 4 and 1\N* 3 is 3. We multiply the
Dialogue: 0,0:01:50.32,0:01:53.21,Default,,0000,0000,0000,,paired elements together and\Nadd the result.
Dialogue: 0,0:01:56.59,0:02:00.78,Default,,0000,0000,0000,,When we come to the first row,\Nsecond column, we work with the
Dialogue: 0,0:02:00.78,0:02:02.71,Default,,0000,0000,0000,,first row here and the second
Dialogue: 0,0:02:02.71,0:02:08.88,Default,,0000,0000,0000,,column here. And again, pairing\Noff 3 * -- 2 is minus 6.
Dialogue: 0,0:02:10.40,0:02:12.19,Default,,0000,0000,0000,,2 * 3.
Dialogue: 0,0:02:13.03,0:02:13.96,Default,,0000,0000,0000,,Is 6.
Dialogue: 0,0:02:15.47,0:02:18.27,Default,,0000,0000,0000,,1 * -- 4 is minus 4.
Dialogue: 0,0:02:19.03,0:02:21.61,Default,,0000,0000,0000,,So in each case, we're\Nmultiplying the paired elements
Dialogue: 0,0:02:21.61,0:02:23.05,Default,,0000,0000,0000,,together and adding the results.
Dialogue: 0,0:02:24.75,0:02:28.94,Default,,0000,0000,0000,,When we want the element that's\Ngoing in here, which is in the
Dialogue: 0,0:02:28.94,0:02:32.80,Default,,0000,0000,0000,,2nd row first column of the\Nanswer, we work with the 2nd
Dialogue: 0,0:02:32.80,0:02:36.66,Default,,0000,0000,0000,,row, first column of the given\Nmatrices 4 * 1 is 4.
Dialogue: 0,0:02:38.22,0:02:41.71,Default,,0000,0000,0000,,Minus 3 * -- 2 is +6.
Dialogue: 0,0:02:43.76,0:02:45.51,Default,,0000,0000,0000,,2 * 3 is 6.
Dialogue: 0,0:02:47.60,0:02:52.03,Default,,0000,0000,0000,,And continuing in the same way,\Nthe answer that goes in the 2nd
Dialogue: 0,0:02:52.03,0:02:55.44,Default,,0000,0000,0000,,row, second column comes from\Ntaking the 2nd row, second
Dialogue: 0,0:02:55.44,0:02:58.85,Default,,0000,0000,0000,,column. 4 * -- 2 is minus 8.
Dialogue: 0,0:03:00.09,0:03:03.50,Default,,0000,0000,0000,,Minus 3 * + 3 is minus 9.
Dialogue: 0,0:03:04.81,0:03:07.40,Default,,0000,0000,0000,,2 * -- 4 is minus 8.
Dialogue: 0,0:03:09.53,0:03:13.89,Default,,0000,0000,0000,,And finally on the last row\Nto find the element in the
Dialogue: 0,0:03:13.89,0:03:17.88,Default,,0000,0000,0000,,first row. Sorry the 3rd row\Nfirst column will work with
Dialogue: 0,0:03:17.88,0:03:21.15,Default,,0000,0000,0000,,the 3rd row, First Column, 5\Nones of five.
Dialogue: 0,0:03:23.17,0:03:25.38,Default,,0000,0000,0000,,4 * -- 2 is minus 8.
Dialogue: 0,0:03:26.41,0:03:28.12,Default,,0000,0000,0000,,3 * 3 is 9.
Dialogue: 0,0:03:29.81,0:03:33.56,Default,,0000,0000,0000,,And similarly to find the last\Nelement, it will be 5 * -- 2,
Dialogue: 0,0:03:33.56,0:03:34.63,Default,,0000,0000,0000,,which is minus 10.
Dialogue: 0,0:03:35.26,0:03:40.75,Default,,0000,0000,0000,,4 * 3 is 12 and 3 * --\N4 is minus 12.
Dialogue: 0,0:03:42.36,0:03:46.40,Default,,0000,0000,0000,,And if we just tidy up what\Nwe've got, we'll have 336
Dialogue: 0,0:03:46.40,0:03:48.09,Default,,0000,0000,0000,,subtract 4, which is 2.
Dialogue: 0,0:03:50.12,0:03:53.22,Default,,0000,0000,0000,,Minus 6 + 6 zero subtract 4 is
Dialogue: 0,0:03:53.22,0:03:57.27,Default,,0000,0000,0000,,minus 4. Four and\Nsix is 10 and 616.
Dialogue: 0,0:03:58.61,0:04:02.91,Default,,0000,0000,0000,,Minus 8 -- 9 --\N8 is minus 25.
Dialogue: 0,0:04:05.45,0:04:07.81,Default,,0000,0000,0000,,5 subtract 8 + 9.
Dialogue: 0,0:04:08.46,0:04:09.53,Default,,0000,0000,0000,,6th
Dialogue: 0,0:04:10.68,0:04:13.45,Default,,0000,0000,0000,,and minus 10 + 12 -- 12 is minus
Dialogue: 0,0:04:13.45,0:04:17.44,Default,,0000,0000,0000,,10. And this is the result\Nof multiplying these two
Dialogue: 0,0:04:17.44,0:04:18.04,Default,,0000,0000,0000,,matrices together.
Dialogue: 0,0:04:20.56,0:04:23.37,Default,,0000,0000,0000,,What about if we try and\Nmultiply the two
Dialogue: 0,0:04:23.37,0:04:25.24,Default,,0000,0000,0000,,matrices together the\Nopposite way round?
Dialogue: 0,0:04:25.24,0:04:27.74,Default,,0000,0000,0000,,Suppose we try and\Nworkout N * M.
Dialogue: 0,0:04:45.73,0:04:50.64,Default,,0000,0000,0000,,Now, in this case the size of\Nthe first matrix here is 3 rows
Dialogue: 0,0:04:50.64,0:04:55.56,Default,,0000,0000,0000,,and two columns, so that's a\Nthree by two and the size of the
Dialogue: 0,0:04:55.56,0:04:59.07,Default,,0000,0000,0000,,second matrix is 3 by 3, three\Nrows, three columns.
Dialogue: 0,0:04:59.89,0:05:03.10,Default,,0000,0000,0000,,And what we observe now is\Nthat these two numbers here
Dialogue: 0,0:05:03.10,0:05:06.61,Default,,0000,0000,0000,,are not the same, they are not\Nequal. That means that we
Dialogue: 0,0:05:06.61,0:05:08.94,Default,,0000,0000,0000,,cannot do the matrix\Nmultiplication in the order
Dialogue: 0,0:05:08.94,0:05:11.57,Default,,0000,0000,0000,,that I've written it down\Nhere. That matrix product
Dialogue: 0,0:05:11.57,0:05:15.07,Default,,0000,0000,0000,,doesn't exist. So this is the\Nfirst point. I'd like to make
Dialogue: 0,0:05:15.07,0:05:18.29,Default,,0000,0000,0000,,that even when you can find a\Nmatrix product by multiplying
Dialogue: 0,0:05:18.29,0:05:20.91,Default,,0000,0000,0000,,two matrices together, it\Nmatters very much. The order
Dialogue: 0,0:05:20.91,0:05:24.42,Default,,0000,0000,0000,,in which you write them down.\NIt may be possible to workout
Dialogue: 0,0:05:24.42,0:05:27.63,Default,,0000,0000,0000,,a product one way, but not\Nanother way. Let's look at
Dialogue: 0,0:05:27.63,0:05:28.51,Default,,0000,0000,0000,,some more examples.
Dialogue: 0,0:05:29.57,0:05:33.12,Default,,0000,0000,0000,,Suppose we've got two matrices\NC&D as I've written them down
Dialogue: 0,0:05:33.12,0:05:37.32,Default,,0000,0000,0000,,here, I'm going to try to work\Nout the product C * D.
Dialogue: 0,0:05:38.12,0:05:41.40,Default,,0000,0000,0000,,And I'll also try and workout\Nthe product D times. See if
Dialogue: 0,0:05:41.40,0:05:42.49,Default,,0000,0000,0000,,either of these exist.
Dialogue: 0,0:05:43.97,0:05:48.39,Default,,0000,0000,0000,,But in the first case, we've got\Na two row three column matrix.
Dialogue: 0,0:05:49.28,0:05:52.21,Default,,0000,0000,0000,,And in the second example\Nhere, within the Second matrix
Dialogue: 0,0:05:52.21,0:05:56.02,Default,,0000,0000,0000,,here we've got three rows into\Ntwo columns, so we can in fact
Dialogue: 0,0:05:56.02,0:05:59.24,Default,,0000,0000,0000,,work this product out because\Nthese numbers are the same and
Dialogue: 0,0:05:59.24,0:06:03.05,Default,,0000,0000,0000,,the result will be a two by\Ntwo matrix. So the shape of
Dialogue: 0,0:06:03.05,0:06:05.69,Default,,0000,0000,0000,,the answer will be 2 rows and\Ntwo columns.
Dialogue: 0,0:06:08.02,0:06:12.61,Default,,0000,0000,0000,,If we try and do this the other\Nway round, D * C, The first
Dialogue: 0,0:06:12.61,0:06:16.59,Default,,0000,0000,0000,,matrix Now has got three rows\Nand two columns. It's a three by
Dialogue: 0,0:06:16.59,0:06:20.26,Default,,0000,0000,0000,,two matrix and the second one's\Ngot two rows and three columns.
Dialogue: 0,0:06:20.26,0:06:22.10,Default,,0000,0000,0000,,It's a two by three matrix.
Dialogue: 0,0:06:22.69,0:06:26.09,Default,,0000,0000,0000,,So you can. You can see that we\Ncan still work it out because
Dialogue: 0,0:06:26.09,0:06:29.25,Default,,0000,0000,0000,,these two numbers are still the\Nsame 2 into the same, but this
Dialogue: 0,0:06:29.25,0:06:32.65,Default,,0000,0000,0000,,time the result is going to be a\Nthree by three matrix, so it's
Dialogue: 0,0:06:32.65,0:06:35.57,Default,,0000,0000,0000,,going to be a bigger matrix with\Nthree rows and three columns.
Dialogue: 0,0:06:38.99,0:06:43.32,Default,,0000,0000,0000,,We can use the process that we\Nevaluate that we worked on
Dialogue: 0,0:06:43.32,0:06:47.29,Default,,0000,0000,0000,,before to evaluate the elements\Nin the these matrices. So for
Dialogue: 0,0:06:47.29,0:06:53.07,Default,,0000,0000,0000,,example, the element that goes\Nin here is 1 * 3 + 2 * 5 added
Dialogue: 0,0:06:53.07,0:06:55.96,Default,,0000,0000,0000,,to 3 * -- 1, which is 10.
Dialogue: 0,0:06:57.73,0:07:02.30,Default,,0000,0000,0000,,And you can check for yourself\Nthat the remaining elements are
Dialogue: 0,0:07:02.30,0:07:03.96,Default,,0000,0000,0000,,131 and minus 11.
Dialogue: 0,0:07:04.78,0:07:08.11,Default,,0000,0000,0000,,So it's possible to workout\NC * D and the answer is a
Dialogue: 0,0:07:08.11,0:07:09.13,Default,,0000,0000,0000,,two by two matrix.
Dialogue: 0,0:07:10.43,0:07:14.85,Default,,0000,0000,0000,,When we do it the other way\Nround, let's take an element
Dialogue: 0,0:07:14.85,0:07:19.26,Default,,0000,0000,0000,,here. Let's take the elements in\Nthe first row, first column and
Dialogue: 0,0:07:19.26,0:07:23.31,Default,,0000,0000,0000,,we obtain the answer by working\Nwith the first row, first
Dialogue: 0,0:07:23.31,0:07:27.73,Default,,0000,0000,0000,,column. Here, that's three\Ntimes, one is 3 added to minus 7
Dialogue: 0,0:07:27.73,0:07:32.14,Default,,0000,0000,0000,,* 4. That's three added to minus\N28, which is minus 25.
Dialogue: 0,0:07:34.76,0:07:39.71,Default,,0000,0000,0000,,And you can proceed in the same\Nway to fill out this resulting
Dialogue: 0,0:07:39.71,0:07:44.67,Default,,0000,0000,0000,,matrix and the numbers. You'll\Nget a -- 25 -- 29 -- 33.
Dialogue: 0,0:07:46.54,0:07:52.62,Default,,0000,0000,0000,,9. 1521\N789
Dialogue: 0,0:07:53.99,0:07:58.54,Default,,0000,0000,0000,,The important point that I want\Nto make here is that when you
Dialogue: 0,0:07:58.54,0:08:00.29,Default,,0000,0000,0000,,multiply C * D together.
Dialogue: 0,0:08:00.95,0:08:06.35,Default,,0000,0000,0000,,It may be possible to also find\ND * C, But the answers that you
Dialogue: 0,0:08:06.35,0:08:09.23,Default,,0000,0000,0000,,get may have completely\Ndifferent sizes. It's certainly
Dialogue: 0,0:08:09.23,0:08:14.27,Default,,0000,0000,0000,,not true that CD is the same as\NDC, so one of the observations
Dialogue: 0,0:08:14.27,0:08:18.95,Default,,0000,0000,0000,,we take away straight away is\Nthat in general CD is not equal
Dialogue: 0,0:08:18.95,0:08:23.27,Default,,0000,0000,0000,,to DC. Even in situations where\Nboth of these products do exist,
Dialogue: 0,0:08:23.27,0:08:25.79,Default,,0000,0000,0000,,we say that matrix\Nmultiplication is not
Dialogue: 0,0:08:25.79,0:08:29.03,Default,,0000,0000,0000,,commutative. In general, it\Nreally doesn't matter the order
Dialogue: 0,0:08:29.03,0:08:31.19,Default,,0000,0000,0000,,in which you carry out the
Dialogue: 0,0:08:31.19,0:08:34.95,Default,,0000,0000,0000,,multiplication. Now that we know\Nhow to multiply 2 matrices
Dialogue: 0,0:08:34.95,0:08:38.05,Default,,0000,0000,0000,,together, I'm going to show you\Nan important property of
Dialogue: 0,0:08:38.05,0:08:43.82,Default,,0000,0000,0000,,identity matrices. Suppose we\Nhave a two by two identity
Dialogue: 0,0:08:43.82,0:08:45.46,Default,,0000,0000,0000,,matrix, that's 1001.
Dialogue: 0,0:08:46.92,0:08:52.25,Default,,0000,0000,0000,,And suppose we have a second\Nmatrix, two 3 -- 4 and seven.
Dialogue: 0,0:08:52.25,0:08:55.94,Default,,0000,0000,0000,,And suppose I want to multiply\Nthese two together.
Dialogue: 0,0:08:58.95,0:09:02.03,Default,,0000,0000,0000,,The identity matrix is\Ncertainly a two by two matrix,
Dialogue: 0,0:09:02.03,0:09:05.73,Default,,0000,0000,0000,,and this matrix is also a two\Nby two matrix. So because
Dialogue: 0,0:09:05.73,0:09:08.81,Default,,0000,0000,0000,,these numbers are the same, we\Ncan actually workout the
Dialogue: 0,0:09:08.81,0:09:12.81,Default,,0000,0000,0000,,product and the answer is also\Na two by two matrix. So the
Dialogue: 0,0:09:12.81,0:09:16.20,Default,,0000,0000,0000,,answer has this sort of shape\Nwith four elements in there.
Dialogue: 0,0:09:18.34,0:09:22.90,Default,,0000,0000,0000,,To get the first element in the\Nanswer, we want to pair 10 with
Dialogue: 0,0:09:22.90,0:09:26.82,Default,,0000,0000,0000,,2 -- 4, multiply the paired\Nelements together and add so we
Dialogue: 0,0:09:26.82,0:09:30.40,Default,,0000,0000,0000,,get 1 * 2 is 2 added to 0 * --
Dialogue: 0,0:09:30.40,0:09:33.18,Default,,0000,0000,0000,,4. Which is just 1 * 2 is 2.
Dialogue: 0,0:09:35.72,0:09:41.08,Default,,0000,0000,0000,,To get this element here, we\Nwant 1 * 3 which is 3 added to 0
Dialogue: 0,0:09:41.08,0:09:43.09,Default,,0000,0000,0000,,* 7, which is just three.
Dialogue: 0,0:09:45.43,0:09:49.63,Default,,0000,0000,0000,,To get the element in here, we\Nwant to pair 01 with two and
Dialogue: 0,0:09:49.63,0:09:54.73,Default,,0000,0000,0000,,minus four, so it's 0 * 2, which\Nis nothing 1 * -- 4 is minus 4,
Dialogue: 0,0:09:54.73,0:09:56.53,Default,,0000,0000,0000,,so we just get minus 4.
Dialogue: 0,0:09:57.63,0:09:59.44,Default,,0000,0000,0000,,And finally, the last element is
Dialogue: 0,0:09:59.44,0:10:03.85,Default,,0000,0000,0000,,0 times. Three, which is nothing\N1 * 7 is 7, so that's our
Dialogue: 0,0:10:03.85,0:10:07.91,Default,,0000,0000,0000,,answer. And if you look at the\Nanswer you'll see the answer is
Dialogue: 0,0:10:07.91,0:10:11.03,Default,,0000,0000,0000,,identical to the matrix we\Nstarted with here. In other
Dialogue: 0,0:10:11.03,0:10:14.15,Default,,0000,0000,0000,,words, multiplying a matrix by\Nan identity matrix when this
Dialogue: 0,0:10:14.15,0:10:16.65,Default,,0000,0000,0000,,multiplication is possible\Nleaves an answer which is
Dialogue: 0,0:10:16.65,0:10:20.08,Default,,0000,0000,0000,,identical to the matrix you\Nstarted with, and that's a very
Dialogue: 0,0:10:20.08,0:10:21.33,Default,,0000,0000,0000,,important property of identity
Dialogue: 0,0:10:21.33,0:10:25.94,Default,,0000,0000,0000,,matrices. The same result occurs\Nif we do the multiplication the
Dialogue: 0,0:10:25.94,0:10:30.86,Default,,0000,0000,0000,,other way round. If we take two\N3 -- 4 seven and we multiply it
Dialogue: 0,0:10:30.86,0:10:34.14,Default,,0000,0000,0000,,by the identity matrix, one\Nnought nought one will find.
Dialogue: 0,0:10:34.14,0:10:38.40,Default,,0000,0000,0000,,It's also possible, and if you\Ngo through the operation 2 * 1
Dialogue: 0,0:10:38.40,0:10:42.34,Default,,0000,0000,0000,,is 2 three times. Nothing is\Nnothing. The result there is 2.
Dialogue: 0,0:10:43.15,0:10:46.00,Default,,0000,0000,0000,,Two times nothing\Nis nothing 313.
Dialogue: 0,0:10:47.44,0:10:51.60,Default,,0000,0000,0000,,Minus 4 * 1 added to 7 times\Nnought is minus 4.
Dialogue: 0,0:10:52.27,0:10:56.13,Default,,0000,0000,0000,,And minus four times North,\Nwhich is nothing added to 717
Dialogue: 0,0:10:56.13,0:11:00.69,Default,,0000,0000,0000,,and you'll see again this answer\Nhere is the same as this matrix
Dialogue: 0,0:11:00.69,0:11:04.20,Default,,0000,0000,0000,,here. So that's very important\Nproperty to remember when you
Dialogue: 0,0:11:04.20,0:11:08.06,Default,,0000,0000,0000,,multiply a matrix by an identity\Nmatrix, it leaves the original
Dialogue: 0,0:11:08.06,0:11:10.87,Default,,0000,0000,0000,,matrix unaltered, identical to\Nwhat it was before.
Dialogue: 0,0:11:11.48,0:11:14.14,Default,,0000,0000,0000,,The same works even if we\Nhaven't got square
Dialogue: 0,0:11:14.14,0:11:16.20,Default,,0000,0000,0000,,matrices. Suppose we have\Nthis identity matrix.
Dialogue: 0,0:11:19.65,0:11:23.18,Default,,0000,0000,0000,,And we multiply, for example by\Nthe Matrix 78.
Dialogue: 0,0:11:24.81,0:11:28.70,Default,,0000,0000,0000,,Well, this has got one row and\Ntwo columns. It's a one by two
Dialogue: 0,0:11:28.70,0:11:32.04,Default,,0000,0000,0000,,matrix. This is got two rows,\Ntwo columns, so we can perform
Dialogue: 0,0:11:32.04,0:11:35.10,Default,,0000,0000,0000,,the matrix multiplication and\Nthe result is going to be a
Dialogue: 0,0:11:35.10,0:11:38.43,Default,,0000,0000,0000,,one by two matrix that's the\Nsame shape as the one we
Dialogue: 0,0:11:38.43,0:11:38.99,Default,,0000,0000,0000,,started with.
Dialogue: 0,0:11:40.36,0:11:43.71,Default,,0000,0000,0000,,And if we carry out the\Noperations, it's 7 * 1, which
Dialogue: 0,0:11:43.71,0:11:46.78,Default,,0000,0000,0000,,is 7 added to 8 times\Nnothing, which is nothing. So
Dialogue: 0,0:11:46.78,0:11:48.17,Default,,0000,0000,0000,,the result is just 7th.
Dialogue: 0,0:11:49.20,0:11:54.02,Default,,0000,0000,0000,,7 times and nothing is nothing\Nand 8 * 1 is 8, so it's just
Dialogue: 0,0:11:54.02,0:11:58.19,Default,,0000,0000,0000,,eight. And again, this answer 7\Neight is the same as the matrix
Dialogue: 0,0:11:58.19,0:12:01.72,Default,,0000,0000,0000,,we started with over here. So\Nthat's just to reinforce the
Dialogue: 0,0:12:01.72,0:12:04.61,Default,,0000,0000,0000,,message that multiplying by an\Nidentity matrix leaves the
Dialogue: 0,0:12:04.61,0:12:05.57,Default,,0000,0000,0000,,original matrix unaltered.