[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:05.13,0:00:08.30,Default,,0000,0000,0000,,One of the most important\Noperations we can do with
Dialogue: 0,0:00:08.30,0:00:11.47,Default,,0000,0000,0000,,matrices is to learn how to\Nmultiply them together. That's
Dialogue: 0,0:00:11.47,0:00:14.96,Default,,0000,0000,0000,,what we're going to do now. And\Nwhen we multiply matrices
Dialogue: 0,0:00:14.96,0:00:18.44,Default,,0000,0000,0000,,together, we find that we\Ncombine the elements in the two
Dialogue: 0,0:00:18.44,0:00:21.93,Default,,0000,0000,0000,,matrices in rather a strange\Nway, and the easiest way to
Dialogue: 0,0:00:21.93,0:00:25.10,Default,,0000,0000,0000,,explain that is by example, so\Nlet's have a look.
Dialogue: 0,0:00:26.04,0:00:29.79,Default,,0000,0000,0000,,Suppose we've got a\Nrow of a matrix 37.
Dialogue: 0,0:00:31.65,0:00:36.06,Default,,0000,0000,0000,,And we want to combine it or\Nmultiply it with a column of
Dialogue: 0,0:00:36.06,0:00:37.07,Default,,0000,0000,0000,,another matrix 29.
Dialogue: 0,0:00:38.79,0:00:42.22,Default,,0000,0000,0000,,And what we do is we combine\Nthese numbers in a rather
Dialogue: 0,0:00:42.22,0:00:46.23,Default,,0000,0000,0000,,strange way. What we do is we\Npair off the elements in the row
Dialogue: 0,0:00:46.23,0:00:49.66,Default,,0000,0000,0000,,of the first matrix with the\Ncolumn with the column in the
Dialogue: 0,0:00:49.66,0:00:52.80,Default,,0000,0000,0000,,Second matrix, and we pair them\Noff and we multiply the
Dialogue: 0,0:00:52.80,0:00:55.66,Default,,0000,0000,0000,,corresponding elements together.\NSo we pair off the three with
Dialogue: 0,0:00:55.66,0:00:58.46,Default,,0000,0000,0000,,the two. The seven with the 9.
Dialogue: 0,0:00:59.25,0:01:02.31,Default,,0000,0000,0000,,And we multiply the paired\Nelements together so we
Dialogue: 0,0:01:02.31,0:01:03.67,Default,,0000,0000,0000,,have 3 * 2.
Dialogue: 0,0:01:06.29,0:01:08.55,Default,,0000,0000,0000,,I mean multiply the\Nseven with the 9.
Dialogue: 0,0:01:10.75,0:01:12.39,Default,,0000,0000,0000,,And we add the results together.
Dialogue: 0,0:01:13.39,0:01:18.54,Default,,0000,0000,0000,,So we have 3 * 2 which is\N6 and 7 * 9, which is 63
Dialogue: 0,0:01:18.54,0:01:22.08,Default,,0000,0000,0000,,and we add them together\Nand we get the answer 69.
Dialogue: 0,0:01:25.56,0:01:29.00,Default,,0000,0000,0000,,So we have this rather strange\Nway in which we have combined
Dialogue: 0,0:01:29.00,0:01:32.45,Default,,0000,0000,0000,,the elements in the row of the\Nfirst matrix with the column
Dialogue: 0,0:01:32.45,0:01:33.60,Default,,0000,0000,0000,,in the Second matrix.
Dialogue: 0,0:01:34.62,0:01:38.65,Default,,0000,0000,0000,,Let's have a look at another\Nexample. Suppose we have a row
Dialogue: 0,0:01:38.65,0:01:39.66,Default,,0000,0000,0000,,which is 425.
Dialogue: 0,0:01:41.02,0:01:45.95,Default,,0000,0000,0000,,And we're going to learn how to\Nmultiply it with a column 368.
Dialogue: 0,0:01:48.67,0:01:53.42,Default,,0000,0000,0000,,And again, what we do is we pair\Nthe elements off elements in the
Dialogue: 0,0:01:53.42,0:01:57.82,Default,,0000,0000,0000,,row of the first matrix with the\Ncolumn of the Second matrix. We
Dialogue: 0,0:01:57.82,0:01:59.18,Default,,0000,0000,0000,,have 4 * 3.
Dialogue: 0,0:02:03.20,0:02:05.06,Default,,0000,0000,0000,,2 * 6.
Dialogue: 0,0:02:07.37,0:02:09.23,Default,,0000,0000,0000,,5 * 8.
Dialogue: 0,0:02:12.05,0:02:17.00,Default,,0000,0000,0000,,And we add these products\Ntogether, so with 4 * 3 which is
Dialogue: 0,0:02:17.00,0:02:23.86,Default,,0000,0000,0000,,12 two times 6 which is 12 and 5\N* 8 which is 40. And if we add
Dialogue: 0,0:02:23.86,0:02:26.91,Default,,0000,0000,0000,,these up will get 12 and 12 is
Dialogue: 0,0:02:26.91,0:02:29.97,Default,,0000,0000,0000,,24. And 40 which is 64.
Dialogue: 0,0:02:34.19,0:02:37.74,Default,,0000,0000,0000,,So this is a rather strange way\Nin which we've combined the
Dialogue: 0,0:02:37.74,0:02:40.100,Default,,0000,0000,0000,,elements in the first matrix\Nwith the elements in the second
Dialogue: 0,0:02:40.100,0:02:43.96,Default,,0000,0000,0000,,matrix, but it's the basis of\Nmatrix multiplication, as we'll
Dialogue: 0,0:02:43.96,0:02:47.21,Default,,0000,0000,0000,,see shortly Now, suppose we have\Nto general matrices A&B, say.
Dialogue: 0,0:02:48.90,0:02:52.05,Default,,0000,0000,0000,,And we want to find the\Nproduct of these two
Dialogue: 0,0:02:52.05,0:02:54.88,Default,,0000,0000,0000,,matrices. In other words, we\Nwant to multiply A&B
Dialogue: 0,0:02:54.88,0:02:55.20,Default,,0000,0000,0000,,together.
Dialogue: 0,0:02:56.47,0:03:00.69,Default,,0000,0000,0000,,Now suppose that this matrix\NA. The first matrix has P,
Dialogue: 0,0:03:00.69,0:03:04.53,Default,,0000,0000,0000,,rossion, Q columns, so it's\Na P by Q matrix.
Dialogue: 0,0:03:06.27,0:03:09.66,Default,,0000,0000,0000,,And the second matrix be.\NLet's suppose that Scott
Dialogue: 0,0:03:09.66,0:03:13.81,Default,,0000,0000,0000,,are rows and S columns,\Nso it's an arby S matrix.
Dialogue: 0,0:03:15.82,0:03:19.75,Default,,0000,0000,0000,,Now it turns out that we can\Nonly form this product. We can
Dialogue: 0,0:03:19.75,0:03:21.26,Default,,0000,0000,0000,,only multiply the two matrices
Dialogue: 0,0:03:21.26,0:03:24.69,Default,,0000,0000,0000,,together. If the number of\Ncolumns in the first matrix,
Dialogue: 0,0:03:24.69,0:03:28.30,Default,,0000,0000,0000,,which is Q is the same as the\Nnumber of rows in the second
Dialogue: 0,0:03:28.30,0:03:31.66,Default,,0000,0000,0000,,matrix, these two numbers have\Ngot to be the same. Q Must equal
Dialogue: 0,0:03:31.66,0:03:35.01,Default,,0000,0000,0000,,R and the reason for that will\Nbecome apparent when we start to
Dialogue: 0,0:03:35.01,0:03:38.37,Default,,0000,0000,0000,,do the calculation. But you've\Ngot to be able to pair up the
Dialogue: 0,0:03:38.37,0:03:41.20,Default,,0000,0000,0000,,elements in the first matrix\Nwith the elements in the Second
Dialogue: 0,0:03:41.20,0:03:44.82,Default,,0000,0000,0000,,matrix and will only be able to\Ndo that if the number of columns
Dialogue: 0,0:03:44.82,0:03:48.43,Default,,0000,0000,0000,,in the first is the same as the\Nnumber of rows in the second.
Dialogue: 0,0:03:49.96,0:03:54.10,Default,,0000,0000,0000,,When that's the case, we can\Nactually find the product AB and
Dialogue: 0,0:03:54.10,0:03:57.90,Default,,0000,0000,0000,,the answer is another matrix.\NAnd let's suppose this answer is
Dialogue: 0,0:03:57.90,0:04:02.38,Default,,0000,0000,0000,,matrix C and the size of matrix.\NSee, we can determine in advance
Dialogue: 0,0:04:02.38,0:04:07.21,Default,,0000,0000,0000,,from the sizes of matrix A&B,\Nthe size of matrix C will be P
Dialogue: 0,0:04:07.21,0:04:11.97,Default,,0000,0000,0000,,by S. So it's got the same\Nnumber of rows as the first
Dialogue: 0,0:04:11.97,0:04:16.09,Default,,0000,0000,0000,,matrix and columns as the second\Nmatrix, so this will be an R.
Dialogue: 0,0:04:16.09,0:04:18.62,Default,,0000,0000,0000,,This will be a P by S matrix.
Dialogue: 0,0:04:20.69,0:04:25.62,Default,,0000,0000,0000,,Let's have a look at a specific\Nexample. Suppose we want to
Dialogue: 0,0:04:25.62,0:04:27.27,Default,,0000,0000,0000,,multiply the matrix 37.
Dialogue: 0,0:04:27.99,0:04:29.67,Default,,0000,0000,0000,,45
Dialogue: 0,0:04:32.04,0:04:34.51,Default,,0000,0000,0000,,by the Matrix 29.
Dialogue: 0,0:04:35.92,0:04:39.08,Default,,0000,0000,0000,,And the first question we should\Nask ourselves is, do these
Dialogue: 0,0:04:39.08,0:04:42.23,Default,,0000,0000,0000,,matrices have the right size so\Nthat we can actually multiply
Dialogue: 0,0:04:42.23,0:04:47.47,Default,,0000,0000,0000,,them together? Well, this matrix\Nis a two row two column matrix.
Dialogue: 0,0:04:50.35,0:04:53.37,Default,,0000,0000,0000,,And the second matrix\Nis 2 rows one column.
Dialogue: 0,0:04:55.96,0:04:59.95,Default,,0000,0000,0000,,And we note that the number of\Ncolumns here in the first matrix
Dialogue: 0,0:04:59.95,0:05:03.94,Default,,0000,0000,0000,,is the same as the number of\Nrows in the second matrix. So
Dialogue: 0,0:05:03.94,0:05:07.63,Default,,0000,0000,0000,,these two numbers are the same,\Nso we can do this multiplication
Dialogue: 0,0:05:07.63,0:05:11.92,Default,,0000,0000,0000,,and the size of the answer. The\Nsize of the result that will get
Dialogue: 0,0:05:11.92,0:05:15.92,Default,,0000,0000,0000,,is obtained by the number of\Nrows in the 1st and columns in
Dialogue: 0,0:05:15.92,0:05:19.91,Default,,0000,0000,0000,,the second. So the size of the\Nanswer that we're looking for is
Dialogue: 0,0:05:19.91,0:05:24.20,Default,,0000,0000,0000,,a 2 by 1 matrix. So you see\Nright at the beginning we can
Dialogue: 0,0:05:24.20,0:05:27.05,Default,,0000,0000,0000,,tell how many. Elements are\Ngoing to be in our answer.
Dialogue: 0,0:05:27.05,0:05:29.77,Default,,0000,0000,0000,,There's going to be a number\Nthere, and a number there so
Dialogue: 0,0:05:29.77,0:05:31.59,Default,,0000,0000,0000,,that we have a 2 by 1 matrix.
Dialogue: 0,0:05:32.92,0:05:36.17,Default,,0000,0000,0000,,Now to determine these\Nnumbers, we use the same
Dialogue: 0,0:05:36.17,0:05:39.78,Default,,0000,0000,0000,,operations as we've just\Nseen. We take the first row
Dialogue: 0,0:05:39.78,0:05:43.75,Default,,0000,0000,0000,,here and we pair the elements\Nwith those in the first
Dialogue: 0,0:05:43.75,0:05:44.11,Default,,0000,0000,0000,,column.
Dialogue: 0,0:05:45.38,0:05:50.31,Default,,0000,0000,0000,,We multiply the paired elements\Ntogether and add the result. So
Dialogue: 0,0:05:50.31,0:05:53.89,Default,,0000,0000,0000,,we want 3 * 2, which is 6.
Dialogue: 0,0:05:55.54,0:06:02.12,Default,,0000,0000,0000,,We want 7 * 9 which is 63 and\Nwe add the results together. 6
Dialogue: 0,0:06:02.12,0:06:07.83,Default,,0000,0000,0000,,and 63 is 69, so the element\Nthat goes in the first position
Dialogue: 0,0:06:07.83,0:06:13.54,Default,,0000,0000,0000,,in our answer is 69. That's 3 *\N2 at 7 * 9.
Dialogue: 0,0:06:15.38,0:06:19.04,Default,,0000,0000,0000,,The element that's going in this\Nposition here is obtained by
Dialogue: 0,0:06:19.04,0:06:22.37,Default,,0000,0000,0000,,working with the 2nd row and\Nthis first column here.
Dialogue: 0,0:06:23.12,0:06:25.87,Default,,0000,0000,0000,,Again, we pair the elements up 4
Dialogue: 0,0:06:25.87,0:06:28.67,Default,,0000,0000,0000,,* 2. Which is 8th.
Dialogue: 0,0:06:29.95,0:06:36.54,Default,,0000,0000,0000,,5 * 9 which is 45, and\Nwe add the results together. 45
Dialogue: 0,0:06:36.54,0:06:38.06,Default,,0000,0000,0000,,+ 8 is.
Dialogue: 0,0:06:39.60,0:06:40.61,Default,,0000,0000,0000,,53
Dialogue: 0,0:06:43.27,0:06:46.81,Default,,0000,0000,0000,,So the result is 6953, so the\Nresult of multiplying these two
Dialogue: 0,0:06:46.81,0:06:50.35,Default,,0000,0000,0000,,matrices together is another\Nmatrix which is a 2 by 1 matrix
Dialogue: 0,0:06:50.35,0:06:53.89,Default,,0000,0000,0000,,and the elements are obtained in\Nthe way I've just shown you.
Dialogue: 0,0:06:54.49,0:06:56.12,Default,,0000,0000,0000,,Let's have a look at another
Dialogue: 0,0:06:56.12,0:07:01.05,Default,,0000,0000,0000,,example. This time I'm going to\Ntry to multiply together the two
Dialogue: 0,0:07:01.05,0:07:06.01,Default,,0000,0000,0000,,matrices A&B where a is this two\Nby two Matrix 2453 and B. Is
Dialogue: 0,0:07:06.01,0:07:09.55,Default,,0000,0000,0000,,this two by two Matrix three 6\N-- 1 nine?
Dialogue: 0,0:07:10.38,0:07:13.34,Default,,0000,0000,0000,,And again, the first question we\Nshould ask ourselves is, do
Dialogue: 0,0:07:13.34,0:07:16.30,Default,,0000,0000,0000,,these matrices have the right\Nsize so that we can actually
Dialogue: 0,0:07:16.30,0:07:21.89,Default,,0000,0000,0000,,multiply them together? Well,\Nmatrix a this matrix A is a two
Dialogue: 0,0:07:21.89,0:07:25.92,Default,,0000,0000,0000,,row two column matrix. So that's\Ntwo by two.
Dialogue: 0,0:07:27.55,0:07:30.96,Default,,0000,0000,0000,,Matrix B is 2 rows and two\Ncolumns, so that's also
Dialogue: 0,0:07:30.96,0:07:31.89,Default,,0000,0000,0000,,two by two.
Dialogue: 0,0:07:33.29,0:07:36.63,Default,,0000,0000,0000,,And you can see that these two\Nnumbers are the same. That is,
Dialogue: 0,0:07:36.63,0:07:39.72,Default,,0000,0000,0000,,the number of columns in the\Nfirst is the same as the
Dialogue: 0,0:07:39.72,0:07:42.80,Default,,0000,0000,0000,,number of rows in the second.\NSo we can perform the matrix
Dialogue: 0,0:07:42.80,0:07:43.06,Default,,0000,0000,0000,,multiplication.
Dialogue: 0,0:07:44.28,0:07:47.60,Default,,0000,0000,0000,,The size of the answer we can\Ndetermine right at the start.
Dialogue: 0,0:07:47.60,0:07:51.20,Default,,0000,0000,0000,,The size of the matrix that we\Nget is determined by the number
Dialogue: 0,0:07:51.20,0:07:54.81,Default,,0000,0000,0000,,here and the number there two by\Ntwo. So what we can decide
Dialogue: 0,0:07:54.81,0:07:58.13,Default,,0000,0000,0000,,before we do any calculation at\Nall is that this answer matrix
Dialogue: 0,0:07:58.13,0:07:59.79,Default,,0000,0000,0000,,is a two by two matrix.
Dialogue: 0,0:08:00.47,0:08:03.20,Default,,0000,0000,0000,,Be 2 rows and two columns,\Nso we're looking for four
Dialogue: 0,0:08:03.20,0:08:04.44,Default,,0000,0000,0000,,numbers to pop in there.
Dialogue: 0,0:08:05.97,0:08:09.39,Default,,0000,0000,0000,,Let's try and figure out how we\Nwork out, what the answer is.
Dialogue: 0,0:08:10.94,0:08:15.27,Default,,0000,0000,0000,,When we want to find the element\Nthat goes in here, observe that
Dialogue: 0,0:08:15.27,0:08:18.60,Default,,0000,0000,0000,,this is the first row first\Ncolumn of the answer.
Dialogue: 0,0:08:20.14,0:08:23.78,Default,,0000,0000,0000,,And the number in the first\Nrow first column comes from
Dialogue: 0,0:08:23.78,0:08:26.76,Default,,0000,0000,0000,,looking at the first row and\N1st Column here.
Dialogue: 0,0:08:28.37,0:08:32.49,Default,,0000,0000,0000,,If we pair off the elements in\Nthe first row and 1st Column
Dialogue: 0,0:08:32.49,0:08:35.03,Default,,0000,0000,0000,,will have 2 * 3 which is 6.
Dialogue: 0,0:08:36.41,0:08:40.96,Default,,0000,0000,0000,,4 * -- 1, which is minus four,\Nand we add them together.
Dialogue: 0,0:08:44.87,0:08:49.43,Default,,0000,0000,0000,,When we come to this element\Nhere, this element is in the
Dialogue: 0,0:08:49.43,0:08:50.95,Default,,0000,0000,0000,,first row, second column.
Dialogue: 0,0:08:51.61,0:08:55.56,Default,,0000,0000,0000,,So we use the first row, second\Ncolumn in the original matrices.
Dialogue: 0,0:08:56.27,0:09:03.16,Default,,0000,0000,0000,,2 * 6 which is 12 and 4\N* 9 Four nines of 36. And
Dialogue: 0,0:09:03.16,0:09:05.91,Default,,0000,0000,0000,,we add those paired\Nproducts together.
Dialogue: 0,0:09:08.61,0:09:13.30,Default,,0000,0000,0000,,When we want the element that's\Nin the 2nd row first column, we
Dialogue: 0,0:09:13.30,0:09:17.100,Default,,0000,0000,0000,,use the 2nd row in the first\Nmatrix and 1st column in the
Dialogue: 0,0:09:17.100,0:09:22.33,Default,,0000,0000,0000,,Second Matrix. Again, pairing\Nthe elements off 5 * 3 is 15.
Dialogue: 0,0:09:23.79,0:09:28.12,Default,,0000,0000,0000,,3 * -- 1 is minus three.\NWhen we add the paired
Dialogue: 0,0:09:28.12,0:09:28.84,Default,,0000,0000,0000,,elements together.
Dialogue: 0,0:09:30.73,0:09:35.57,Default,,0000,0000,0000,,Finally. The element that's in\Nthe 2nd row, second column of
Dialogue: 0,0:09:35.57,0:09:39.86,Default,,0000,0000,0000,,the answer is obtained by using\Nthe elements in the 2nd row of
Dialogue: 0,0:09:39.86,0:09:42.83,Default,,0000,0000,0000,,the first matrix, second column\Nof the Second matrix.
Dialogue: 0,0:09:43.81,0:09:50.98,Default,,0000,0000,0000,,5 * 6 which is 30 and 3 *\N9, which is 27, and we add the
Dialogue: 0,0:09:50.98,0:09:52.25,Default,,0000,0000,0000,,paired elements together.
Dialogue: 0,0:09:53.48,0:09:57.60,Default,,0000,0000,0000,,So finally, just to tidy it\Nup, we've got 6 subtract 4
Dialogue: 0,0:09:57.60,0:09:58.62,Default,,0000,0000,0000,,which is 2.
Dialogue: 0,0:10:00.54,0:10:03.61,Default,,0000,0000,0000,,12 + 36, which is 48.
Dialogue: 0,0:10:05.75,0:10:12.53,Default,,0000,0000,0000,,15 subtract 3 which is 12 and 30\N+ 27 which is 57, so we can find
Dialogue: 0,0:10:12.53,0:10:18.12,Default,,0000,0000,0000,,the matrix product AB in this\Ncase and the result is a two by
Dialogue: 0,0:10:18.12,0:10:21.87,Default,,0000,0000,0000,,two matrix. Let's have a look\Nat another example. Suppose
Dialogue: 0,0:10:21.87,0:10:24.65,Default,,0000,0000,0000,,we're asked to find the\Nproduct of these two matrices.
Dialogue: 0,0:10:25.99,0:10:29.27,Default,,0000,0000,0000,,And again, we should ask all\Nthese matrices of the
Dialogue: 0,0:10:29.27,0:10:32.55,Default,,0000,0000,0000,,appropriate size. The first\Nmatrix here has two rows, one
Dialogue: 0,0:10:32.55,0:10:34.85,Default,,0000,0000,0000,,column, it's a 2 by 1 matrix.
Dialogue: 0,0:10:35.76,0:10:40.15,Default,,0000,0000,0000,,And the second matrix has two\Nrows and two columns, so it's a
Dialogue: 0,0:10:40.15,0:10:44.21,Default,,0000,0000,0000,,two by two matrix. Now in this\Ncase, you'll see that the
Dialogue: 0,0:10:44.21,0:10:48.60,Default,,0000,0000,0000,,number of columns in the first\Nmatrix is not the same as the
Dialogue: 0,0:10:48.60,0:10:52.32,Default,,0000,0000,0000,,number of rows in the second\Nmatrix. Those two numbers are
Dialogue: 0,0:10:52.32,0:10:55.03,Default,,0000,0000,0000,,not equal, so we cannot\Nmultiply these matrices
Dialogue: 0,0:10:55.03,0:10:58.41,Default,,0000,0000,0000,,together. We say the product of\Nthese matrices doesn't exist,
Dialogue: 0,0:10:58.41,0:11:01.11,Default,,0000,0000,0000,,so we stop there. We can't\Ncalculate that.
Dialogue: 0,0:11:02.54,0:11:07.95,Default,,0000,0000,0000,,Let's have another example.\NSuppose we want to try to
Dialogue: 0,0:11:07.95,0:11:11.74,Default,,0000,0000,0000,,find the product of the\Nmatrices 3214.
Dialogue: 0,0:11:14.18,0:11:17.70,Default,,0000,0000,0000,,With the matrix XY. Now this\Nis the first example we've
Dialogue: 0,0:11:17.70,0:11:20.90,Default,,0000,0000,0000,,looked at where we've had\Nsymbols rather than numbers in
Dialogue: 0,0:11:20.90,0:11:23.78,Default,,0000,0000,0000,,our matrix, but the operation\Nthe process is, the
Dialogue: 0,0:11:23.78,0:11:25.38,Default,,0000,0000,0000,,calculations are just the\Nsame.
Dialogue: 0,0:11:26.43,0:11:29.47,Default,,0000,0000,0000,,First of all, we should ask can\Nwe multiply these together? Are
Dialogue: 0,0:11:29.47,0:11:30.73,Default,,0000,0000,0000,,they of the right size?
Dialogue: 0,0:11:31.66,0:11:34.19,Default,,0000,0000,0000,,This is a two row two\Ncolumn matrix.
Dialogue: 0,0:11:35.97,0:11:38.56,Default,,0000,0000,0000,,And this is a two row one\Ncolumn matrix.
Dialogue: 0,0:11:40.03,0:11:43.97,Default,,0000,0000,0000,,And these numbers are the same\Nin here the number of columns in
Dialogue: 0,0:11:43.97,0:11:48.21,Default,,0000,0000,0000,,the first is the same as the\Nnumber of rows in the second. So
Dialogue: 0,0:11:48.21,0:11:50.94,Default,,0000,0000,0000,,we can actually perform the\Nmatrix multiplication and the
Dialogue: 0,0:11:50.94,0:11:55.79,Default,,0000,0000,0000,,answer we get will be a 2 by 1\Nmatrix, so we know the shape of
Dialogue: 0,0:11:55.79,0:11:59.72,Default,,0000,0000,0000,,the answer. It's a two row one\Ncolumn matrix, so it looks the
Dialogue: 0,0:11:59.72,0:12:01.85,Default,,0000,0000,0000,,same shape as this one. This one
Dialogue: 0,0:12:01.85,0:12:04.35,Default,,0000,0000,0000,,here. Two rows, one column.
Dialogue: 0,0:12:06.34,0:12:09.75,Default,,0000,0000,0000,,Let's actually work out what the\Nelements in the answers are.
Dialogue: 0,0:12:10.60,0:12:15.18,Default,,0000,0000,0000,,As before, we take the first row\Nand pair the elements with the
Dialogue: 0,0:12:15.18,0:12:21.62,Default,,0000,0000,0000,,first column. So it's 3\Nmultiplied by X 2 * y and we add
Dialogue: 0,0:12:21.62,0:12:26.13,Default,,0000,0000,0000,,the resulting products, so we\Nget three X + 2 Y.
Dialogue: 0,0:12:33.19,0:12:37.14,Default,,0000,0000,0000,,The element that's down here,\Nwhich is in the 2nd row, first
Dialogue: 0,0:12:37.14,0:12:41.09,Default,,0000,0000,0000,,column of our answer, is\Nobtained by using the 2nd row in
Dialogue: 0,0:12:41.09,0:12:43.06,Default,,0000,0000,0000,,the first matrix and the first
Dialogue: 0,0:12:43.06,0:12:47.61,Default,,0000,0000,0000,,column here. Multiply the pad\Nelements together and add so
Dialogue: 0,0:12:47.61,0:12:54.34,Default,,0000,0000,0000,,it's 1 * X which is X 4 * y,\Nwhich is 4 Y and we add the
Dialogue: 0,0:12:54.34,0:13:00.33,Default,,0000,0000,0000,,products together and we get X +\N4 Y. So the result we found is a
Dialogue: 0,0:13:00.33,0:13:02.20,Default,,0000,0000,0000,,two row one column matrix.
Dialogue: 0,0:13:03.77,0:13:06.91,Default,,0000,0000,0000,,In this case, the answers got\Nsymbols in as well, but that's
Dialogue: 0,0:13:06.91,0:13:09.53,Default,,0000,0000,0000,,the result of finding the\Nproduct of these two matrices.
Dialogue: 0,0:13:09.53,0:13:12.94,Default,,0000,0000,0000,,Now we can go on and look at\Nmore examples and trying to
Dialogue: 0,0:13:12.94,0:13:15.30,Default,,0000,0000,0000,,find products of matrices of\Ndifferent sizes and shapes,
Dialogue: 0,0:13:15.30,0:13:18.18,Default,,0000,0000,0000,,and we'll do some more of that\Nin the next video.