0:00:00.000,0:00:00.670


0:00:00.670,0:00:02.380
Well, after the last video,[br]hopefully, we're a little

0:00:02.380,0:00:04.330
familiar with how you[br]add matrices.

0:00:04.330,0:00:07.150
So now let's learn how[br]to multiply matrices.

0:00:07.150,0:00:11.450
And keep in mind, these are[br]human-created definitions for

0:00:11.450,0:00:12.840
matrix multiplication.

0:00:12.840,0:00:15.000
We could have come up with[br]completely different ways to

0:00:15.000,0:00:15.450
multiply it.

0:00:15.450,0:00:18.510
But I encourage you to learn[br]this way because it'll help

0:00:18.510,0:00:19.810
you in math class.

0:00:19.810,0:00:22.100
And we will see later that[br]there's actually a lot of

0:00:22.100,0:00:24.730
applications that come out[br]of this type of matrix

0:00:24.730,0:00:25.290
multiplication.

0:00:25.290,0:00:26.340
So let me think of[br]two matrices.

0:00:26.340,0:00:30.320
I will do two 2 by 2 matrices,[br]and let's multiply them.

0:00:30.320,0:00:34.500
Let's say-- let me pick some[br]random numbers: 2,

0:00:34.500,0:00:40.530
minus 3, 7, and 5.

0:00:40.530,0:00:42.980
And I'm going to multiply that[br]matrix, or that table of

0:00:42.980,0:00:56.440
numbers, times 10, minus 8--[br]let me pick a good number

0:00:56.440,0:01:03.980
here-- 12, and then minus 2.

0:01:03.980,0:01:07.400
So now there might be a strong[br]temptation-- and you know in

0:01:07.400,0:01:11.330
some ways it's not even an[br]illegitimate temptation-- to

0:01:11.330,0:01:13.880
do the same thing with[br]multiplication that we did

0:01:13.880,0:01:17.520
with addition, to just multiply[br]the corresponding

0:01:17.520,0:01:20.580
terms. So you might be tempted[br]to say, well, the first term

0:01:20.580,0:01:23.160
right here, the 1, 1 term, or[br]in the first row and first

0:01:23.160,0:01:25.070
column, is going to[br]be 2 times 10.

0:01:25.070,0:01:26.970
And this term is going[br]to be minus 3 times

0:01:26.970,0:01:28.230
minus 8 and so forth.

0:01:28.230,0:01:30.330
And that's how we added matrices[br]so maybe it's a

0:01:30.330,0:01:33.510
natural extension to multiply[br]matrices the same way.

0:01:33.510,0:01:36.180
And that is legitimate.

0:01:36.180,0:01:38.530
One could define it that way,[br]but that's not the way it is

0:01:38.530,0:01:39.410
in the real world.

0:01:39.410,0:01:40.380
And the way in the real world,

0:01:40.380,0:01:42.470
unfortunately, is more complex.

0:01:42.470,0:01:44.980
But if you look at a[br]bunch of examples I

0:01:44.980,0:01:45.670
think you'll get it.

0:01:45.670,0:01:47.460
And you'll learn that[br]it's actually fairly

0:01:47.460,0:01:47.980
straightforward.

0:01:47.980,0:01:48.985
So how do we do it?

0:01:48.985,0:01:53.290
So this first term that's in[br]the first row and its first

0:01:53.290,0:01:58.050
column, is equal to essentially[br]this first row's

0:01:58.050,0:02:01.380
vector-- no, this first[br]row vector--

0:02:01.380,0:02:04.790
times this column vector.

0:02:04.790,0:02:08.229
Now what do I mean[br]by that, right?

0:02:08.229,0:02:11.320
So it's getting it's row[br]information from the first

0:02:11.320,0:02:14.170
matrix's row, and it's getting[br]it's column information from

0:02:14.170,0:02:16.320
the second matrix's column.

0:02:16.320,0:02:17.190
So how do I do that?

0:02:17.190,0:02:18.860
If you're familiar with dot[br]product, it's essentially the

0:02:18.860,0:02:20.930
dot product of these[br]two matrices.

0:02:20.930,0:02:24.820
Or without saying it that fancy,[br]it's just this: it's 2

0:02:24.820,0:02:31.890
times 10, so 2-- I'm going to[br]write small-- times 10, plus

0:02:31.890,0:02:39.640
minus 3 times 12.

0:02:39.640,0:02:42.630
I'm going to run out of space.

0:02:42.630,0:02:45.600
And so what's this second[br]term over here?

0:02:45.600,0:02:49.170
Well, we're still on the first[br]row of the product vector but

0:02:49.170,0:02:50.350
now we're on the[br]second column.

0:02:50.350,0:02:51.910
We get our column information[br]from here.

0:02:51.910,0:02:58.800
So let's pick a good color--[br]this is a slightly different

0:02:58.800,0:03:00.530
shade of purple.

0:03:00.530,0:03:04.110
So now this is going to be--[br]I'll do that in another

0:03:04.110,0:03:10.580
color-- 2 times minus 8-- let me[br]just write out the number--

0:03:10.580,0:03:18.810
2 times minus 8 is minus 16,[br]plus minus 3 times minus 2--

0:03:18.810,0:03:21.160
what's minus 3 times minus 2?

0:03:21.160,0:03:26.400
That is plus 6, right?

0:03:26.400,0:03:28.870
So that's in row 1 column 2.

0:03:28.870,0:03:30.590
It's minus 16 plus 6.

0:03:30.590,0:03:31.830
And then let's come down here.

0:03:31.830,0:03:33.920
So now we're in the[br]second row.

0:03:33.920,0:03:35.550
So now we're going to use--[br]we're getting our row

0:03:35.550,0:03:37.930
information from the first[br]matrix-- I know this is

0:03:37.930,0:03:41.380
confusing and I feel bad for you[br]right now, but we're going

0:03:41.380,0:03:45.230
to a bunch of examples and[br]I think it'll make sense.

0:03:45.230,0:03:49.060
So this term-- the bottom left[br]term-- is going to be this row

0:03:49.060,0:03:50.310
times this column.

0:03:50.310,0:03:58.010
So it's going to be 7 times[br]10, so 70, plus 7 times 10

0:03:58.010,0:04:05.650
plus 5 times 12, plus 60.

0:04:05.650,0:04:09.290
And then the bottom right term[br]is going to be 7 times minus

0:04:09.290,0:04:19.540
8, which is minus 56 plus[br]5 times minus 2.

0:04:19.540,0:04:22.900
So that's minus 10.

0:04:22.900,0:04:31.290
So the final product is going to[br]be 2 times 10 is 20, minus

0:04:31.290,0:04:41.040
36, so that's minus 16[br]plus 6, that's 10.

0:04:41.040,0:04:42.220
90-- was that what I said?

0:04:42.220,0:04:46.390
No, it was-- 70, plus[br]60, that's 130.

0:04:46.390,0:04:55.580
And then minus 56 minus[br]10, so minus 66.

0:04:55.580,0:04:56.400
So there you have it.

0:04:56.400,0:04:59.020
We just multiplied this matrix[br]times this matrix.

0:04:59.020,0:05:00.360
Let me do another example.

0:05:00.360,0:05:03.490
And I think I'll actually[br]squeeze it on this side so

0:05:03.490,0:05:06.106
that we can write this side out[br]a little bit more neatly.

0:05:06.106,0:05:18.630
So let's take the matrix and[br]now 1, 2, 3, 4, times the

0:05:18.630,0:05:27.710
matrix 5, 6, 7, 8.

0:05:27.710,0:05:30.200
Now we have much more space to[br]work with so this should come

0:05:30.200,0:05:32.760
out neater.

0:05:32.760,0:05:37.460
OK, but I'm going to do the[br]same thing, so to get this

0:05:37.460,0:05:40.350
term right here-- the top left[br]term-- we're going to take--

0:05:40.350,0:05:43.070
or the one that has row 1 column[br]1-- we're going to take

0:05:43.070,0:05:52.050
the row 1 information from[br]here, and the column 1

0:05:52.050,0:05:54.440
information from here.

0:05:54.440,0:05:56.070
So you can view it as[br]this row vector

0:05:56.070,0:05:57.120
times this column vector.

0:05:57.120,0:06:01.680
So it results, 1 times[br]5 plus 2 times 7.

0:06:01.680,0:06:07.730


0:06:07.730,0:06:08.380
Right?

0:06:08.380,0:06:09.610
There you go.

0:06:09.610,0:06:12.520
And so this term, it'll be this[br]row vector times this

0:06:12.520,0:06:15.740
column vector-- let me do that[br]in a different color-- will be

0:06:15.740,0:06:20.550
1 times 6 plus 2 times 8.

0:06:20.550,0:06:21.370
Let me write that down.

0:06:21.370,0:06:28.590
So it's 1 times 6[br]plus 2 times 8.

0:06:28.590,0:06:33.270


0:06:33.270,0:06:34.720
Now we go down to[br]the second row.

0:06:34.720,0:06:38.130
And we get our row information[br]from the first vector-- let me

0:06:38.130,0:06:44.150
circle it with this color--[br]and it is 3 times 5

0:06:44.150,0:06:45.520
plus 4 times 7.

0:06:45.520,0:06:52.800


0:06:52.800,0:06:54.760
And then we are in the bottom[br]right, so we're in the bottom

0:06:54.760,0:06:57.070
row and second column.

0:06:57.070,0:06:59.350
So we get our row information[br]from here and our column

0:06:59.350,0:07:00.930
information from here.

0:07:00.930,0:07:03.860
So it's 3 times 6[br]plus 4 times 8.

0:07:03.860,0:07:10.300


0:07:10.300,0:07:13.680
And if we simplify,[br]this is 5 plus--

0:07:13.680,0:07:15.950
Well actually, let me just[br]remind you where all the

0:07:15.950,0:07:16.630
numbers came from.

0:07:16.630,0:07:18.200
So we have that green[br]color, right?

0:07:18.200,0:07:25.890
This 1 and this 2, that's[br]this 1 and this 2,

0:07:25.890,0:07:28.620
this 1 and this 2.

0:07:28.620,0:07:28.755
Right?

0:07:28.755,0:07:30.330
And notice, these were in the[br]first row and they're in the

0:07:30.330,0:07:31.660
first row here.

0:07:31.660,0:07:33.640
And this 5 and this 7?

0:07:33.640,0:07:39.920
Well, that's this 5 and this[br]7, and this 5 and this 7.

0:07:39.920,0:07:42.800
So, interesting.

0:07:42.800,0:07:45.410
This was in column 1 of the[br]second matrix and this is in

0:07:45.410,0:07:47.780
column 1 in the product[br]matrix.

0:07:47.780,0:07:51.030
And similarly, the[br]6 and the 8.

0:07:51.030,0:07:55.610
That's this 6, this 8, and then[br]it's used here, this 6

0:07:55.610,0:07:57.540
and this 8.

0:07:57.540,0:08:00.250
And then finally this 3 and the[br]4 in the brown, so that's

0:08:00.250,0:08:03.980
this 3, this 4, and[br]this 3 and this 4.

0:08:03.980,0:08:05.290
And we could of course[br]simplify all of it.

0:08:05.290,0:08:10.180
This was 1 times 5 plus 2 times[br]7, so that's 5 plus 14,

0:08:10.180,0:08:15.410
so this is 19.

0:08:15.410,0:08:19.200
This is 1 times 6 plus 2[br]times 8, so it's 6 plus

0:08:19.200,0:08:22.330
16, so that's 22.

0:08:22.330,0:08:26.380
This is 3 times 5[br]plus 4 times 7.

0:08:26.380,0:08:32.909
So 15 plus 28, 38, 43-- if my[br]math is correct-- and then we

0:08:32.909,0:08:36.250
have 3 times 6 plus 4 times 8.

0:08:36.250,0:08:44.220
So that's 18 plus[br]32, that's 50.

0:08:44.220,0:08:45.980
So now let me ask you-- just so[br]you know that the product

0:08:45.980,0:08:47.690
matrix-- just write[br]it neatly-- is

0:08:47.690,0:08:53.610
19, 22, 43, and 50.

0:08:53.610,0:08:55.100
So now let me ask[br]you a question.

0:08:55.100,0:08:58.810
When we did matrix addition we[br]learned that if I had two

0:08:58.810,0:09:03.210
matrices-- it didn't matter what[br]order we added them in.

0:09:03.210,0:09:06.970
So if we said, A plus B-- and[br]these are matrices; that's why

0:09:06.970,0:09:09.340
I'm making them all bold-- we[br]said this is the same thing as

0:09:09.340,0:09:12.330
B plus A, based on how[br]we define matrix

0:09:12.330,0:09:15.520
addition, B plus A.

0:09:15.520,0:09:17.060
So now let me ask[br]you a question.

0:09:17.060,0:09:22.970
Is multiplying two matrices,[br]is AB-- that's just means

0:09:22.970,0:09:26.460
we're multiplying A and B-- is[br]that the same thing as BA?

0:09:26.460,0:09:30.090


0:09:30.090,0:09:30.820
Does it matter?

0:09:30.820,0:09:33.800
Does the order of the matrix[br]multiplication matter?

0:09:33.800,0:09:36.310
And so, I'll tell you right[br]now, it actually matters a

0:09:36.310,0:09:37.090
tremendous amount.

0:09:37.090,0:09:39.550
And actually there are certain[br]matrices that you can add in

0:09:39.550,0:09:42.040
one direction that you can't[br]add in the other-- oh, that

0:09:42.040,0:09:47.300
you can multiply in one way but[br]you can't multiply in the

0:09:47.300,0:09:48.200
other order.

0:09:48.200,0:09:50.755
And well, I'll show you that[br]in an example-- but just to

0:09:50.755,0:09:52.800
show that this isn't even equal[br]for most matrices, I

0:09:52.800,0:09:56.640
encourage you to multiply these[br]two matrices in the

0:09:56.640,0:09:57.110
other order.

0:09:57.110,0:09:58.950
Actually let me do that.

0:09:58.950,0:10:00.630
Let me do that really[br]fast just to prove

0:10:00.630,0:10:01.440
the point to you.

0:10:01.440,0:10:02.830
So let me delete all[br]this top part.

0:10:02.830,0:10:06.040


0:10:06.040,0:10:13.830
Let me delete all of it, and[br]actually I can delete to this.

0:10:13.830,0:10:15.960
So hopefully, you know that when[br]I multiply this matrix

0:10:15.960,0:10:17.850
times this matrix, I got this.

0:10:17.850,0:10:20.250
So let me switch the order--[br]and I'll do it fairly fast

0:10:20.250,0:10:23.140
just so as to not bore you-- so[br]let me switch the order of

0:10:23.140,0:10:24.150
the matrix multiplication.

0:10:24.150,0:10:26.755
This is good as this is another[br]example-- so I'm going

0:10:26.755,0:10:36.740
to multiply this matrix: 5, 6,[br]7, 8, times this matrix-- and

0:10:36.740,0:10:40.400
I just switched the order; and[br]we're testing to see whether

0:10:40.400,0:10:46.700
order matters-- 1, 2, 3, 4.

0:10:46.700,0:10:49.010
Let's do it-- and I won't do all[br]the colors and everything,

0:10:49.010,0:10:49.690
I'll just do it systematically.

0:10:49.690,0:10:54.330
I think you just have to see a[br]lot of examples here-- So this

0:10:54.330,0:10:56.410
first term gets its row[br]information from the first

0:10:56.410,0:10:58.810
matrix, column information[br]from the second matrix.

0:10:58.810,0:11:06.340
So it's 5 times 1 plus 6 times[br]3, so it's 5 times 1--

0:11:06.340,0:11:09.130
Let me just write,[br]actually edit.

0:11:09.130,0:11:17.650
I'm going to skip a step here--[br]OK so it's 5 times 1

0:11:17.650,0:11:23.480
plus 6 times 3, plus 18.

0:11:23.480,0:11:25.110
What's the second term here?

0:11:25.110,0:11:29.650
It's going to be 5 times[br]2 plus 6 times 4.

0:11:29.650,0:11:40.190
So 5 times 2 is 10, plus[br]6 times 4 is 24.

0:11:40.190,0:11:42.390
Right, now we just took[br]this row times this

0:11:42.390,0:11:44.680
column right here.

0:11:44.680,0:11:48.030
OK now we're down here for the[br]set-- so then we're doing this

0:11:48.030,0:11:51.095
row, this element right here at[br]the bottom left is going to

0:11:51.095,0:11:52.960
use this row, and this column.

0:11:52.960,0:12:00.410
So it's 7 times 1[br]plus 8 times 3.

0:12:00.410,0:12:02.980
8 times 3 is 24.

0:12:02.980,0:12:05.430
And then finally, to get this[br]element we're essentially

0:12:05.430,0:12:11.820
multiplying this row times this[br]column, so it's 7 times 2

0:12:11.820,0:12:21.810
is 14, plus 8 times[br]4, plus 32.

0:12:21.810,0:12:30.020
So this is equal to 5[br]plus 18 is 23, 34.

0:12:30.020,0:12:31.100
What's 7 plus 24?

0:12:31.100,0:12:35.530
That's 31, 46.

0:12:35.530,0:12:44.140
So notice, if we called[br]this matrix A and this

0:12:44.140,0:12:47.080
is matrix B, right?

0:12:47.080,0:12:57.560
In the last example, we showed[br]that A times B is equal to 19,

0:12:57.560,0:13:02.640
22, 43, 50.

0:13:02.640,0:13:06.550
And we just showed that, well,[br]if you reverse the order, B

0:13:06.550,0:13:10.225
times A is actually this[br]completely different matrix.

0:13:10.225,0:13:12.230
So the order in which[br]you multiply

0:13:12.230,0:13:15.140
matrices completely matters.

0:13:15.140,0:13:16.330
So I'm actually running[br]out of time.

0:13:16.330,0:13:19.210
In the next video I going talk[br]a little bit more about the

0:13:19.210,0:13:22.260
types of matrix-- well, one, we[br]know that order matters--

0:13:22.260,0:13:25.110
and in the next video I'll[br]show that what type of

0:13:25.110,0:13:27.460
matrices can be multiplied[br]by each other.

0:13:27.460,0:13:29.820
When we added or subtracted[br]matrices, we just said, well

0:13:29.820,0:13:31.850
they have to have the same[br]dimensions because you're

0:13:31.850,0:13:33.855
adding or subtracting[br]corresponding terms. But

0:13:33.855,0:13:37.040
you'll see with multiplication[br]it's a little bit different.

0:13:37.040,0:13:38.253
And we'll do that in[br]the next video.

0:13:38.253,0:13:39.790
See you soon.

0:13:39.790,0:13:39.900