Determinant as scaling factor

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:03.250
Let me graph our coordinates
in R2.

00:00:03.250 --> 00:00:06.512
So let's see, let me do
it fairly neatly.

00:00:06.512 --> 00:00:09.160
I want to do it neater
than that.

00:00:09.160 --> 00:00:13.990
So that is my vertical
coordinate, this is my

00:00:13.990 --> 00:00:16.149
horizontal coordinate.

00:00:16.149 --> 00:00:16.640
Good enough.

00:00:16.640 --> 00:00:20.450
So let's say I have four points
in R2 specified by four

00:00:20.450 --> 00:00:21.445
position vectors.

00:00:21.445 --> 00:00:23.390
So this is R2 right here.

00:00:23.390 --> 00:00:26.990
So my first position vector
is the zero vector.

00:00:26.990 --> 00:00:31.050
So it just specifies the point
right there, so it's 0, 0.

00:00:31.050 --> 00:00:40.720
My second point is specified the
by the vector say k 1, 0,

00:00:40.720 --> 00:00:42.720
so some constant times 0.

00:00:42.720 --> 00:00:44.210
Let me write it a little
bit neater than that,

00:00:44.210 --> 00:00:46.020
a little bit bigger.

00:00:46.020 --> 00:00:50.960
So it's k 1, 0, so that's my
second position vector, which

00:00:50.960 --> 00:00:53.220
would specify that point
right there.

00:00:53.220 --> 00:00:56.010
So let me draw my vector
right there.

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Let's say my third point
is going to be up here.

00:01:01.190 --> 00:01:03.672
As you can, see I'm creating
a rectangle.

00:01:03.672 --> 00:01:06.720
Let's say it's specified
by the position vector.

00:01:06.720 --> 00:01:09.190
Let me draw the position
vector like that.

00:01:09.190 --> 00:01:13.700
And that position vector is
going to be the vector k1.

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That's going to be its
horizontal component and then

00:01:17.020 --> 00:01:20.480
k2 is its vertical component.

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If we wanted to graph it on the
coordinates, this would be

00:01:23.005 --> 00:01:25.140
the k2 coordinate right there.

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And let's say that the last
point is this point right

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here, and it is specified by
the position vector 0, k2.

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And let me define a set.

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Let me call that set

00:01:43.770 --> 00:01:45.955
rectangle, say R for rectangle.

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Let me just call it Rec.

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Let's say that rectangle is
equal to the set of all of the

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points when you connect the
points specified by those

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position vectors.

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So the rectangle formed by
connecting points specified by

00:02:17.580 --> 00:02:18.440
these guys here.

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So if we call this vector a,
this is vector b, this is

00:02:23.760 --> 00:02:28.750
vector c, and then this is
vector d right here, so

00:02:28.750 --> 00:02:31.090
rectangle specified
by connecting the

00:02:31.090 --> 00:02:38.590
points a, b, c and d.

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Now let's say-- so let me draw
what our set rectangle is

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going to be.

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It's going to be all of these
points here, all of these

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points there, all of those
points there, and then all of

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those points there, including
our vertices.

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So that is my rectangle.

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Now, what is the area within
this rectangle?

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So this is the area
of rectangle.

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What's the area within
this rectangle?

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Well, it's just base
times height.

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The base has length k1.

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The distance from here to here
is k1, so it's just k1.

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And what's the height?

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It's the distance from here
to here, which is just k2.

00:03:22.470 --> 00:03:25.820
So it's just k1 times k2.

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That's fairly straightforward.

00:03:27.310 --> 00:03:30.390
I haven't shown you anything
that fascinating just yet.

00:03:30.390 --> 00:03:33.280
But let's say we're going
to transform this set.

00:03:33.280 --> 00:03:35.540
We're going to transform this
rectangle and we're going to

00:03:35.540 --> 00:03:39.870
transform it with the
transformation T.

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Let me pick a nice
color for T.

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Let's say we have the
transformation T.

00:03:44.760 --> 00:03:50.540
It is a mapping from R2 to R2,
and it can be represented as--

00:03:50.540 --> 00:03:53.100
so when you apply the
transformation T to some

00:03:53.100 --> 00:04:01.960
vector x, it is equal to the
matrix a, b, c, d times your

00:04:01.960 --> 00:04:04.410
vector x, just like that.

00:04:04.410 --> 00:04:06.440
Now, let's see what happens
when we apply the

00:04:06.440 --> 00:04:08.560
transformation to each
of these points.

00:04:08.560 --> 00:04:12.140
We saw many videos ago that if
you want essentially the image

00:04:12.140 --> 00:04:15.880
of our rectangle under this
transformation, you can

00:04:15.880 --> 00:04:17.860
essentially just get the
transformation of each of the

00:04:17.860 --> 00:04:21.200
end points and then connect
the dots in your codomain,

00:04:21.200 --> 00:04:22.610
which is also going to be R2.

00:04:22.610 --> 00:04:24.820
So we just have to figure out
the transformation of each of

00:04:24.820 --> 00:04:26.020
these guys.

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So what is my transformation?

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I'll do it over here.

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I'll do it in the colors.

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What is the transformation
applied to the zero vector,

00:04:35.910 --> 00:04:38.270
applied to point
a right there?

00:04:38.270 --> 00:04:43.860
Well, it's just going to be
a times 0 plus b times 0,

00:04:43.860 --> 00:04:46.050
which is just 0.

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And then the second term is
going to be c times 0 plus d

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times 0, which is just 0.

00:04:52.010 --> 00:04:54.530
That's vector a.

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Vector b.

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If I apply the transformation
to vector b, vector b is k1,

00:05:02.100 --> 00:05:03.740
0, just like that.

00:05:03.740 --> 00:05:07.210
If I apply the transformation,
it's going to be a times k--

00:05:07.210 --> 00:05:11.610
so the first entry in the
transformed vector or what

00:05:11.610 --> 00:05:13.880
we're mapping to in our codomain
is going to be a

00:05:13.880 --> 00:05:19.300
times k1 plus b times
0, so it's just ak1.

00:05:19.300 --> 00:05:21.980
And then the second entry's
going to be c times k1

00:05:21.980 --> 00:05:24.400
plus d times 0.

00:05:24.400 --> 00:05:27.390
So it's going to
be c times k1.

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All I'm doing is taking the
matrix vector product of this

00:05:29.900 --> 00:05:31.970
guy and that guy.

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And then let's move
on to point c.

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So the transformation of point
c of k1, k2 is equal to-- or

00:05:45.330 --> 00:05:46.910
the transformation of
vector c, we could

00:05:46.910 --> 00:05:48.760
say, is equal to what?

00:05:48.760 --> 00:05:52.980
The first entry's going to be
a times k1 plus b times k2.

00:05:52.980 --> 00:05:57.695
a times k1 plus b times k2, and
the second entry's going

00:05:57.695 --> 00:06:00.270
to be c times k1 plus
d times k2.

00:06:04.660 --> 00:06:07.220
We have one point left.

00:06:07.220 --> 00:06:10.310
The transformation, if we take
the transformation of our

00:06:10.310 --> 00:06:15.730
vector 0, k2, what do we get?

00:06:15.730 --> 00:06:22.620
We get a times 0 plus b times
k2, which is just b times k2,

00:06:22.620 --> 00:06:26.900
and then c times 0 plus d times
k2, which is just d

00:06:26.900 --> 00:06:29.780
times k2, just like that.

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So let's draw the image of
our rectangle under our

00:06:33.260 --> 00:06:33.735
transformation.

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So let me redraw my axes.

00:06:36.760 --> 00:06:40.120
So this is my vertical
axis right there.

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This is my horizontal
axis right there.

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And so a gets mapped to the
zero vector right there.

00:06:48.250 --> 00:06:54.440
So this is the transformation
of 0, 0 right like that.

00:06:54.440 --> 00:06:57.400
Now, what does the vector
b get transformed?

00:06:57.400 --> 00:07:01.480
The vector b gets transformed
to ak1, ck1, so let me

00:07:01.480 --> 00:07:03.230
draw it like this.

00:07:03.230 --> 00:07:05.930
The vector b gets transformed
to something that looks like

00:07:05.930 --> 00:07:09.130
this, where this is the vector--
so this is the

00:07:09.130 --> 00:07:20.110
transformation of k1 and 0,
which is equal to ak1, ck1.

00:07:20.110 --> 00:07:21.400
We saw that already.

00:07:21.400 --> 00:07:24.240
Now, what does this yellow
guy get transformed to?

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Let me do the blue one first.
So the blue one gets

00:07:26.490 --> 00:07:28.390
transformed here.

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Let me finish here.

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So if we come down here,
this is going to

00:07:33.300 --> 00:07:35.340
be ak1 right there.

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If we go to the left like that,
that's going to ck1.

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That's how we graph it.

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Now this last guy, this blue
guy, maybe he looks

00:07:43.980 --> 00:07:46.870
something like this.

00:07:49.500 --> 00:07:53.640
This is the transformation of
that vector right there by

00:07:53.640 --> 00:08:00.930
this linear transformation, 0,
k2, which is equal to the

00:08:00.930 --> 00:08:10.420
vector bk2 and dk2, so this
point right here is bk2 and

00:08:10.420 --> 00:08:14.320
then this coordinate
right there is dk2.

00:08:14.320 --> 00:08:17.660
Now this last guy, this yellow
point, when I transformed it,

00:08:17.660 --> 00:08:18.860
what do I get?

00:08:18.860 --> 00:08:23.700
I get ak1, plus bk2, so ak1 plus
bk2 will get you right

00:08:23.700 --> 00:08:26.620
about there, so that's going
to be its x-coordinate.

00:08:26.620 --> 00:08:28.630
And then its y-coordinate
is ck1 plus dk2.

00:08:31.810 --> 00:08:37.559
So ck1, and then I'm going to
add that much again to it.

00:08:37.559 --> 00:08:40.250
So it's going to get me
someplace over here.

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So I'm going to add this
distance to this whole

00:08:42.530 --> 00:08:44.400
distance over here, so
I'm going to get

00:08:44.400 --> 00:08:45.670
up here some point.

00:08:45.670 --> 00:08:48.620
I'm going to get right
over here.

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So that vector's going to look
something like that.

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That is the transformation
of k1, k2.

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That's the transformation
of that guy.

00:09:00.400 --> 00:09:02.180
And then if we connect the
dots, remember, the

00:09:02.180 --> 00:09:05.860
transformation of this set of
this rectangle, or we could

00:09:05.860 --> 00:09:08.000
say the image of this
rectangle under the

00:09:08.000 --> 00:09:10.890
transformation, we just take
each of the points that define

00:09:10.890 --> 00:09:13.310
that rectangle and then
we connect the dots.

00:09:13.310 --> 00:09:14.950
We saw that awhile ago.

00:09:14.950 --> 00:09:19.000
So let me draw it this way.

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This line right here that
connects those two dots will

00:09:22.840 --> 00:09:26.240
be transformed to this line that
connects these two dots

00:09:26.240 --> 00:09:27.950
right there.

00:09:27.950 --> 00:09:32.170
This line-- I was going to do
that in a different color.

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This line that connects those
two dots is going to be

00:09:34.640 --> 00:09:38.600
transformed to this line that
connects those two dots.

00:09:38.600 --> 00:09:41.990
And then we have this line that
connects those two dots

00:09:41.990 --> 00:09:45.230
is going to be transformed to
this line that connects these

00:09:45.230 --> 00:09:46.740
two dots right there.

00:09:46.740 --> 00:09:49.610
And then finally, this line that
connects these two dots

00:09:49.610 --> 00:09:52.090
is going to be transformed to
this line that connects these

00:09:52.090 --> 00:09:53.800
two dots right there.

00:09:53.800 --> 00:09:57.220
Now, just out of curiosity,
what is-- let

00:09:57.220 --> 00:09:58.380
me just write this.

00:09:58.380 --> 00:10:01.510
So this rectangle right
here is the image.

00:10:01.510 --> 00:10:06.580
We could write this as T of our
rectangle, or if we wanted

00:10:06.580 --> 00:10:11.220
to write it in words, this is
equal to the image of the

00:10:11.220 --> 00:10:19.670
rectangle under our
transformation T.

00:10:19.670 --> 00:10:27.880
Now, what is the area of the
image of our rectangle under

00:10:27.880 --> 00:10:29.570
our transformation?

00:10:29.570 --> 00:10:31.800
What is the area
of this thing?

00:10:31.800 --> 00:10:35.670
What is this area right there?

00:10:35.670 --> 00:10:39.670
Well, we could view this
parallelogram essentially as a

00:10:39.670 --> 00:10:45.590
parallelogram generated by this
vector and that vector.

00:10:45.590 --> 00:10:48.360
Or if we wanted to write it
another way, if we had a

00:10:48.360 --> 00:10:52.950
matrix, if we had some matrix
whose column vectors are this

00:10:52.950 --> 00:10:56.915
guy and that guy, so let's say,
it's first column vector

00:10:56.915 --> 00:11:03.140
is this guy, so ak1, ck1, And
this one right here, this was

00:11:03.140 --> 00:11:07.780
the transformation of k1, 0.

00:11:07.780 --> 00:11:12.210
And then its second column
vector is this guy right here,

00:11:12.210 --> 00:11:18.810
which was bk2 and dk2.

00:11:18.810 --> 00:11:24.330
This parallelogram is a
parallelogram generated by

00:11:24.330 --> 00:11:25.700
these two column vectors.

00:11:25.700 --> 00:11:27.610
This was the transformation
right here.

00:11:27.610 --> 00:11:33.300
This was the transformation
of 0, k2, right?

00:11:33.300 --> 00:11:37.670
And in the last video, we said
that the area of this

00:11:37.670 --> 00:11:41.450
parallelogram is equivalent
to-- let me

00:11:41.450 --> 00:11:42.650
write this right now.

00:11:42.650 --> 00:11:44.220
So the area of the
parallelogram, which is the

00:11:44.220 --> 00:11:46.970
same thing as the area of the
image of our original

00:11:46.970 --> 00:11:49.820
rectangle under our
transformation, the area this

00:11:49.820 --> 00:11:54.220
parallelogram is equivalent to
the absolute value-- let me

00:11:54.220 --> 00:11:56.230
call this determinant
right here, let me

00:11:56.230 --> 00:11:58.580
call this I for image.

00:11:58.580 --> 00:12:07.420
So it's equal to the absolute
value of the determinant of

00:12:07.420 --> 00:12:11.910
the matrix whose column
vectors generate that

00:12:11.910 --> 00:12:13.010
parallelogram.

00:12:13.010 --> 00:12:16.800
So it should be equal to the
determinant of the absolute

00:12:16.800 --> 00:12:18.810
value of the determinant of I.

00:12:18.810 --> 00:12:20.200
We saw that in the last video.

00:12:20.200 --> 00:12:22.720
That's what the area of this
thing is going to be.

00:12:22.720 --> 00:12:27.190
Well, what is the determinant
of this?

00:12:27.190 --> 00:12:29.126
Well, this is going to be equal
to-- we've got to keep

00:12:29.126 --> 00:12:30.990
our absolute value signs.

00:12:30.990 --> 00:12:35.970
So the determinant of this is
ak1 times dk2, so we could

00:12:35.970 --> 00:12:38.480
right this as-- let
me switch colors.

00:12:38.480 --> 00:12:44.840
We could write it
as k1, k2, ad.

00:12:44.840 --> 00:12:46.910
All I did is multiply that term
times that term minus

00:12:46.910 --> 00:12:48.750
this term times that term.

00:12:48.750 --> 00:12:51.430
So minus k1, k2, bc.

00:12:55.280 --> 00:12:56.930
That's just the definition
of the

00:12:56.930 --> 00:12:59.170
determinant of a 2-by-2 matrix.

00:12:59.170 --> 00:13:00.470
We saw this in the last video.

00:13:00.470 --> 00:13:03.050
This is nothing new here, or
maybe it might be relatively

00:13:03.050 --> 00:13:05.590
new if you just watched the last
video, but all I'm saying

00:13:05.590 --> 00:13:07.700
is, look, if we want to figure
out this parallelogram, this

00:13:07.700 --> 00:13:10.030
is the parallelogram
generated by that

00:13:10.030 --> 00:13:11.560
vector and that vector.

00:13:11.560 --> 00:13:13.840
And if we say that those two
guys are the column vectors of

00:13:13.840 --> 00:13:17.990
some matrix right there, we saw
on the last video that the

00:13:17.990 --> 00:13:21.140
area of this parallelogram is
equal to the absolute value of

00:13:21.140 --> 00:13:23.000
the determinant of
this matrix.

00:13:23.000 --> 00:13:24.710
Now, what is the equal to?

00:13:24.710 --> 00:13:26.950
This is equal to-- we can
factor out a k1, k2.

00:13:26.950 --> 00:13:37.220
So it's the absolute value of
k1, k2 times ad minus bc.

00:13:37.220 --> 00:13:39.410
Now what is this equal to?

00:13:39.410 --> 00:13:45.530
Well, ad minus bc, that is
the determinant of our

00:13:45.530 --> 00:13:47.590
transformation vector.

00:13:47.590 --> 00:13:50.980
So if we say that T of x-- if we
called that vector-- or our

00:13:50.980 --> 00:13:52.610
transformation matrix.

00:13:52.610 --> 00:13:58.900
If we call this matrix A, if we
call it A, this is equal to

00:13:58.900 --> 00:14:05.070
k1, k2 times the determinant
of A, which is a pretty

00:14:05.070 --> 00:14:06.480
interesting takeaway.

00:14:06.480 --> 00:14:11.020
What this is telling us is,
look, if I have some region,

00:14:11.020 --> 00:14:13.230
or in this case, I just have the
rectangle, but if I have

00:14:13.230 --> 00:14:17.360
some region in my domain and I
apply a transformation, and

00:14:17.360 --> 00:14:19.390
let's say my region
has some area.

00:14:21.960 --> 00:14:24.460
Let's say in this case,
it has some area A.

00:14:24.460 --> 00:14:29.480
Let me write this
right as area.

00:14:29.480 --> 00:14:33.290
This was my original area in
my original region, right?

00:14:33.290 --> 00:14:35.760
And then I apply some
transformation to it where the

00:14:35.760 --> 00:14:41.760
transformation is equal to some
matrix A times any member

00:14:41.760 --> 00:14:43.020
of your domain.

00:14:43.020 --> 00:14:44.710
I apply some transformation
to it.

00:14:44.710 --> 00:14:46.520
I get some new region.

00:14:46.520 --> 00:14:51.170
I get the image of this set
under my transformation.

00:14:51.170 --> 00:14:57.150
Your new area is going to be
equal to the absolute value of

00:14:57.150 --> 00:15:00.980
the determinant of your
transformation matrix.

00:15:00.980 --> 00:15:03.800
That's the determinant of the
transformation matrix.

00:15:03.800 --> 00:15:05.365
That's the transformation
matrix.

00:15:10.520 --> 00:15:12.050
So it's equal to the
determinant of the

00:15:12.050 --> 00:15:17.730
transformation matrix times
the area of your original

00:15:17.730 --> 00:15:22.460
rectangle in this case, right?

00:15:22.460 --> 00:15:23.990
This was your original
rectangle.

00:15:23.990 --> 00:15:26.500
You take the absolute value,
just because sometimes you

00:15:26.500 --> 00:15:31.540
swap these vectors you might
get a negative sign, so you

00:15:31.540 --> 00:15:32.500
take the absolute value.

00:15:32.500 --> 00:15:34.510
But that's a really neat idea.

00:15:34.510 --> 00:15:38.630
The determinant of the
transformation matrix is

00:15:38.630 --> 00:15:43.020
essentially a scaling
factor on the area

00:15:43.020 --> 00:15:44.420
of a certain region.

00:15:44.420 --> 00:15:46.600
Now, I'm not going to prove it
to you here, but you can kind

00:15:46.600 --> 00:15:49.100
of imagine it.

00:15:49.100 --> 00:15:52.620
Let's say I have some-- let
me go abstract now.

00:15:52.620 --> 00:15:57.960
Let's say I have some region in
R2-- let me do a better--

00:15:57.960 --> 00:16:00.600
let me do a shape that
you might recognize.

00:16:00.600 --> 00:16:05.970
Let's say I have something
like that.

00:16:05.970 --> 00:16:07.960
Let's say it's an
ellipse, right?

00:16:07.960 --> 00:16:09.210
And this is our domain.

00:16:11.580 --> 00:16:13.530
It's R2.

00:16:13.530 --> 00:16:16.750
And let's say that the area
here, the area of this region

00:16:16.750 --> 00:16:19.570
right here, is equal to A.

00:16:19.570 --> 00:16:23.330
So area is equal to A.

00:16:23.330 --> 00:16:27.180
And let's say I have some
transformation T that is a

00:16:27.180 --> 00:16:35.800
mapping from R2 to R2 And T is
defined, so T applied to some

00:16:35.800 --> 00:16:39.485
vector x in my domain is equal
to-- well, I already used A,

00:16:39.485 --> 00:16:45.130
so let's say it's equal to some
matrix B times any vector

00:16:45.130 --> 00:16:46.430
in my domain.

00:16:46.430 --> 00:16:50.210
If I take the image of this area
under T, so this is my

00:16:50.210 --> 00:16:51.190
domain right here.

00:16:51.190 --> 00:16:52.300
Let me do it in the codomain.

00:16:52.300 --> 00:16:54.850
Let's say the image of this
under B, let's say it looks

00:16:54.850 --> 00:16:57.860
something like-- or my image
under T-- let's say it looks

00:16:57.860 --> 00:16:58.940
something like this.

00:16:58.940 --> 00:17:00.550
I don't know what it's going to
look like, but let's say it

00:17:00.550 --> 00:17:01.800
looks something like this.

00:17:04.339 --> 00:17:11.060
Let's say it looks something
like that right there, so this

00:17:11.060 --> 00:17:12.490
right here.

00:17:12.490 --> 00:17:16.460
So if we call this set-- let
me call it E for ellipse.

00:17:16.460 --> 00:17:17.839
That's this whole thing
right here.

00:17:17.839 --> 00:17:20.280
This is the ellipse
right there.

00:17:20.280 --> 00:17:24.140
This is the image of my
ellipse under the

00:17:24.140 --> 00:17:25.170
transformation T.

00:17:25.170 --> 00:17:27.440
If I take every point of this
ellipse, I'll construct this

00:17:27.440 --> 00:17:28.260
right here.

00:17:28.260 --> 00:17:32.910
Or I guess maybe just to keep
the analogy straight for this

00:17:32.910 --> 00:17:36.050
example, let's say that the
ellipse is the set of just the

00:17:36.050 --> 00:17:39.820
boundary, but it also applies
actually if you were to fill

00:17:39.820 --> 00:17:41.090
up the whole region.

00:17:41.090 --> 00:17:46.450
So this is boundary gets
mapped to this

00:17:46.450 --> 00:17:47.380
boundary right here.

00:17:47.380 --> 00:17:48.820
But it would have worked
just as well if we

00:17:48.820 --> 00:17:49.640
filled up the region.

00:17:49.640 --> 00:17:52.900
Since I'm not proving either to
you very rigorously, you'll

00:17:52.900 --> 00:17:54.570
have to accept it as
a leap of faith.

00:17:54.570 --> 00:17:58.650
So this is the image of this
boundary under our

00:17:58.650 --> 00:17:59.800
transformation.

00:17:59.800 --> 00:18:02.650
And if the area surrounded by
this boundary, or if you could

00:18:02.650 --> 00:18:07.570
say the area of this region is
A, then the area here, the

00:18:07.570 --> 00:18:13.790
area of the image of our
ellipse under our

00:18:13.790 --> 00:18:17.750
transformation is going to be
equal to the absolute value of

00:18:17.750 --> 00:18:22.090
our area, our original area,
right there times the

00:18:22.090 --> 00:18:27.565
determinant of our
transformation matrix.

00:18:31.310 --> 00:18:34.580
If you find it a little bit of
a big leap to go from just an

00:18:34.580 --> 00:18:37.310
arbitrary rectangle
transformation to maybe these

00:18:37.310 --> 00:18:41.090
more generalized shapes, you
can imagine-- so T is going

00:18:41.090 --> 00:18:42.230
from there to there.

00:18:42.230 --> 00:18:44.300
It's always nice to
draw that arrow.

00:18:44.300 --> 00:18:48.830
But if you view it as a bit of
a leap to make that jump from

00:18:48.830 --> 00:18:52.530
rectangles to curves, you can
imagine this shape to be able

00:18:52.530 --> 00:18:58.020
to be constructed by a bunch
of arbitrary rectangles.

00:18:58.020 --> 00:19:01.320
So you could essentially fill
this space with a bunch of

00:19:01.320 --> 00:19:09.120
arbitrary rectangles like
this, and you could keep

00:19:09.120 --> 00:19:11.970
making them infinitely
small so that you

00:19:11.970 --> 00:19:15.000
completely fill the space.

00:19:15.000 --> 00:19:16.730
You completely fill that
space with rectangles.

00:19:16.730 --> 00:19:20.180
And if you were to map each of
those rectangles using T, each

00:19:20.180 --> 00:19:23.560
of those rectangles might look
something like this.

00:19:23.560 --> 00:19:25.350
Maybe they look something
like this.

00:19:25.350 --> 00:19:27.760
I don't know what they'll
look like.

00:19:27.760 --> 00:19:32.970
They'll look like a bunch of
parallelograms. Maybe they'll

00:19:32.970 --> 00:19:35.620
look something like that.

00:19:35.620 --> 00:19:37.820
I'm not drawing it
that neatly.

00:19:37.820 --> 00:19:41.140
They'll look like a bunch of
parallelograms, and they will

00:19:41.140 --> 00:19:43.040
essentially fill this
space right there.

00:19:43.040 --> 00:19:46.090
So that's a way you can get your
head around the idea of

00:19:46.090 --> 00:19:49.200
jumping from just the arbitrary
rectangles to these

00:19:49.200 --> 00:19:51.500
arbitrary curves or
shapes or regions.

00:19:51.500 --> 00:19:53.310
But this is a pretty neat
outcome, and it's a very

00:19:53.310 --> 00:19:55.850
interesting way to view
a determinant.

00:19:55.850 --> 00:19:58.100
A determinant of a
transformation matrix is

00:19:58.100 --> 00:20:02.190
essentially a scaling factor for
area as you map from one

00:20:02.190 --> 00:20:05.560
region to another region, or
as we go from one region to

00:20:05.560 --> 00:20:08.590
the image of that region under
the transformation.


Determinant and area of a parallelogram

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:04.800
I've got a 2 by 2 matrix here,
and let's just say its entries

00:00:04.800 --> 00:00:08.690
are a, b, c, and d.

00:00:08.690 --> 00:00:11.140
And it's composed of
two column vectors.

00:00:11.140 --> 00:00:15.430
We've done this before, let's
call this first column v1 and

00:00:15.430 --> 00:00:19.440
let's call the second
column v2.

00:00:19.440 --> 00:00:20.860
So we can rewrite here.

00:00:20.860 --> 00:00:28.440
We can say v1 one is equal to
ac, and we could write that v2

00:00:28.440 --> 00:00:31.890
is equal to bd.

00:00:31.890 --> 00:00:35.270
And these are both members of
r2, and just to have a nice

00:00:35.270 --> 00:00:38.870
visualization in our head,
let's graph these two.

00:00:38.870 --> 00:00:41.170
Let me draw my axes.

00:00:41.170 --> 00:00:43.870
That's my vertical axis.

00:00:43.870 --> 00:00:45.870
That's my horizontal axis.

00:00:45.870 --> 00:00:48.240
And maybe v1 looks something
like this.

00:00:48.240 --> 00:00:50.860
v1 might look something
like that.

00:00:50.860 --> 00:00:53.040
So that is v1.

00:00:53.040 --> 00:00:57.222
It's horizontal component will
be a, its vertical coordinant

00:00:57.222 --> 00:01:00.870
-- give you this as maybe a
position vector, or just how

00:01:00.870 --> 00:01:02.950
we're drawing it, is c.

00:01:02.950 --> 00:01:06.190
And then v2, let's just say it
looks something like this.

00:01:06.190 --> 00:01:08.380
Let's say that they're
not the same vector.

00:01:08.380 --> 00:01:10.380
So v2 looks like that.

00:01:10.380 --> 00:01:15.230
v2, its horizontal coordinate
is going to b, and its

00:01:15.230 --> 00:01:18.790
vertical coordinate
is going to be d.

00:01:18.790 --> 00:01:21.830
Now, what we're going to concern
ourselves with in this

00:01:21.830 --> 00:01:25.190
video is the parallelogram
generated by these two guys.

00:01:25.190 --> 00:01:28.770
What I mean by that is, imagine
that these two guys

00:01:28.770 --> 00:01:31.480
are position vectors that are
specifying points on a

00:01:31.480 --> 00:01:35.160
parallelogram, and then of
course the -- or not of course

00:01:35.160 --> 00:01:37.990
but, the origin is also
another point in the

00:01:37.990 --> 00:01:39.760
parallelogram, so what will
be the last point on the

00:01:39.760 --> 00:01:40.970
parallelogram?

00:01:40.970 --> 00:01:42.110
Well, you can imagine.

00:01:42.110 --> 00:01:45.380
A parallelogram, we already have
two sides of it, so the

00:01:45.380 --> 00:01:46.770
other two sides have
to be parallel.

00:01:46.770 --> 00:01:51.130
So one side look like that,
parallel to v1 the way I've

00:01:51.130 --> 00:01:54.700
drawn it, and the other side
will look like this.

00:01:54.700 --> 00:01:56.030
That's our parallelogram.

00:01:56.030 --> 00:02:00.410
The parallelogram generated
by v2 and v1.

00:02:00.410 --> 00:02:02.700
What we're going to concern
ourselves with specifically is

00:02:02.700 --> 00:02:14.150
the area of the parallelogram
generated by v1 and v2.

00:02:14.150 --> 00:02:16.120
So how do we figure that out?

00:02:16.120 --> 00:02:18.510
In general, if I have just any
parallelogram-- this is kind

00:02:18.510 --> 00:02:21.056
of a tilted one, but if I just
have any parallelogram, let me

00:02:21.056 --> 00:02:26.060
just draw any parallelogram
right there-- the area is just

00:02:26.060 --> 00:02:29.060
equal to the base-- so
that could be the

00:02:29.060 --> 00:02:32.060
base-- times the height.

00:02:32.060 --> 00:02:34.945
So it's equal to base -- I'll
write capital B since we have

00:02:34.945 --> 00:02:38.210
a lowercase b there--
base times height.

00:02:38.210 --> 00:02:40.560
That's what the area of a
parallelogram would be.

00:02:40.560 --> 00:02:44.040
Now what are the base and the
height in this situation?

00:02:44.040 --> 00:02:45.640
Let me write this down.

00:02:45.640 --> 00:02:48.850
The area of our parallelogram
is equal to the

00:02:48.850 --> 00:02:51.310
base times the height.

00:02:51.310 --> 00:02:53.250
And actually-- well, let
me just write it here.

00:02:53.250 --> 00:02:55.010
So what is the base here?

00:02:55.010 --> 00:03:00.380
The base here is going to be
the length of our vector v.

00:03:00.380 --> 00:03:02.730
So this is our base.

00:03:02.730 --> 00:03:08.020
So this right here is going to
be the length of vector v1,

00:03:08.020 --> 00:03:10.470
the length of this orange
vector right here.

00:03:10.470 --> 00:03:14.340
And what's the height of this
parallelogram going to be?

00:03:14.340 --> 00:03:19.660
We could drop a perpendicular
here, and that, the length of

00:03:19.660 --> 00:03:23.086
this line right here, is
going to be our height.

00:03:25.710 --> 00:03:29.240
So how can we figure out that,
you know, we know what v1 is,

00:03:29.240 --> 00:03:31.010
so we can figure out the
base pretty easily.

00:03:31.010 --> 00:03:35.490
But how can we figure
out the height?

00:03:35.490 --> 00:03:39.110
Well, one thing we can do is, if
we can figure out this guy

00:03:39.110 --> 00:03:43.580
right here, we could use the
Pythagorean theorem.

00:03:43.580 --> 00:03:49.110
Because the length of this
vector squared, plus H

00:03:49.110 --> 00:03:53.130
squared, is going to be equal
to the length of v2 squared.

00:03:53.130 --> 00:03:54.960
So let's see if we
can do that.

00:03:54.960 --> 00:03:56.080
What is this guy?

00:03:56.080 --> 00:03:58.510
What is this green
guy right here?

00:03:58.510 --> 00:04:03.160
Well if you imagine a line--
let's imagine some line l.

00:04:03.160 --> 00:04:09.410
So let's say l is a line
spanned by v1.

00:04:15.390 --> 00:04:20.279
Which means you take all of the
multiples of v1, and all

00:04:20.279 --> 00:04:22.580
of the positions that they
specify will create a set of

00:04:22.580 --> 00:04:24.345
points, and that is my line l.

00:04:24.345 --> 00:04:26.530
So you take all the multiples
of v1, you're going to get

00:04:26.530 --> 00:04:29.835
every point along this line.

00:04:29.835 --> 00:04:31.950
Just like that.

00:04:31.950 --> 00:04:35.714
Now if we have l defined that
way-- that line right there is

00:04:35.714 --> 00:04:36.610
l, I don't know if
you can see it.

00:04:36.610 --> 00:04:38.680
Let me do it a little bit better
-- and it goes through

00:04:38.680 --> 00:04:40.560
v1 and it just keeps
going over there.

00:04:40.560 --> 00:04:43.840
What is this green
line right there?

00:04:43.840 --> 00:04:46.450
This green line that we're
concerned with, that's the

00:04:46.450 --> 00:04:52.080
projection onto l of what?

00:04:52.080 --> 00:04:54.830
Well, we have a perpendicular
here, you can imagine the

00:04:54.830 --> 00:04:57.060
light source coming down-- I
don't know if that analogy

00:04:57.060 --> 00:04:58.540
helps you-- but it's kind
of the shadow of

00:04:58.540 --> 00:05:00.850
v2 onto that line.

00:05:00.850 --> 00:05:06.490
So it's a projection of v2, of
your vector v2 onto l is this

00:05:06.490 --> 00:05:08.360
green line right there.

00:05:08.360 --> 00:05:13.010
So, if we want to figure out
H, we can just use the

00:05:13.010 --> 00:05:14.550
Pythagorean theorem.

00:05:14.550 --> 00:05:19.830
So we can say that the length
of H squared-- well I'm just

00:05:19.830 --> 00:05:21.370
writing H as the length,
I'm not even

00:05:21.370 --> 00:05:23.040
specifying it as a vector.

00:05:23.040 --> 00:05:26.560
So we could say that H squared,
where that is the

00:05:26.560 --> 00:05:31.550
length of this line, plus the
length of this vector

00:05:31.550 --> 00:05:34.130
squared-- and the length of
that vector squared is the

00:05:34.130 --> 00:05:37.010
length of the projection
onto l of v2.

00:05:37.010 --> 00:05:40.130
I'll do that in a
different color.

00:05:40.130 --> 00:05:48.720
So the length of the projection
onto l of v2

00:05:48.720 --> 00:05:49.870
squared-- all right?

00:05:49.870 --> 00:05:51.530
We're just doing the Pythagorean
theorem.

00:05:51.530 --> 00:05:52.870
This squared plus this
squared is going

00:05:52.870 --> 00:05:54.320
to equal that squared.

00:05:54.320 --> 00:06:01.930
It's going to be equal to the
length of v2 squared.

00:06:01.930 --> 00:06:03.410
That's just the Pythagorean
theorem.

00:06:07.670 --> 00:06:10.550
Nothing fancy there.

00:06:10.550 --> 00:06:11.740
So how can we simplify?

00:06:11.740 --> 00:06:14.470
We want to solve for H.

00:06:14.470 --> 00:06:17.230
And actually, let's just solve
for H squared for now because

00:06:17.230 --> 00:06:19.200
it'll keep things a little
bit simpler.

00:06:19.200 --> 00:06:25.160
So we can say that H squared is
equal to this guy, is equal

00:06:25.160 --> 00:06:30.400
to the length of my vector v2
squared minus the length of

00:06:30.400 --> 00:06:32.390
the projection squared.

00:06:32.390 --> 00:06:36.740
So minus -- I'll do that in
purple -- minus the length of

00:06:36.740 --> 00:06:42.410
the projection onto
l of v2 squared.

00:06:42.410 --> 00:06:44.750
Remember, this thing is just
this thing right here, we're

00:06:44.750 --> 00:06:46.860
just doing the Pythagorean
theorem.

00:06:46.860 --> 00:06:49.680
So let's see if we can simplify
this, or write it in

00:06:49.680 --> 00:06:51.700
terms that we understand.

00:06:51.700 --> 00:06:55.620
So the length of a vector
squared, this is just equal

00:06:55.620 --> 00:06:58.390
to-- let me write it this
way-- this is just

00:06:58.390 --> 00:07:02.630
equal to v2 dot v2.

00:07:02.630 --> 00:07:04.880
You take a vector, you dot it
with itself, and you get the

00:07:04.880 --> 00:07:06.550
length of that vector
squared, we saw that

00:07:06.550 --> 00:07:08.370
many, many videos ago.

00:07:08.370 --> 00:07:11.710
And then what is this guy
going to be equal to?

00:07:11.710 --> 00:07:13.410
Well, the projection--
I'll do it over here.

00:07:13.410 --> 00:07:20.700
The projection onto l of v2 is
going to be equal to v2 dot

00:07:20.700 --> 00:07:23.580
the spanning vector,
which is v1.

00:07:23.580 --> 00:07:31.200
So it's v2 dot v1 over the
spanning vector dotted with

00:07:31.200 --> 00:07:33.810
itself, v1 dot v1.

00:07:33.810 --> 00:07:36.370
We saw this several videos
ago when we learned about

00:07:36.370 --> 00:07:37.380
projections.

00:07:37.380 --> 00:07:40.510
And this is just a number
right there.

00:07:40.510 --> 00:07:42.860
And then it's going
to be times the

00:07:42.860 --> 00:07:44.960
spanning vector itself.

00:07:44.960 --> 00:07:48.400
So times v1.

00:07:48.400 --> 00:07:50.390
That is what the
projection is.

00:07:50.390 --> 00:07:55.000
So it's going to be this
minus the length of

00:07:55.000 --> 00:07:56.440
the projection squared.

00:07:56.440 --> 00:07:59.100
What is the length of the
projection squared?

00:07:59.100 --> 00:08:04.080
Well that's this guy dotted
with himself.

00:08:04.080 --> 00:08:07.970
Let me write it this way, let
me take it step by step.

00:08:07.970 --> 00:08:12.240
So this is going to be minus--
I'm want to make sure I can

00:08:12.240 --> 00:08:16.250
still see that up there so I
don't have to rewrite it.

00:08:16.250 --> 00:08:19.110
The projection is going to be,
it looks a little complicated

00:08:19.110 --> 00:08:25.020
but hopefully things will
simplify, v2 dot v1 over v1

00:08:25.020 --> 00:08:31.250
dot v1 times-- switch colors--
times the vector-- this is all

00:08:31.250 --> 00:08:33.370
just going to end up being a
number, remember you take dot

00:08:33.370 --> 00:08:37.050
products, you get numbers--
times the vector v1.

00:08:37.050 --> 00:08:38.570
And we're going to take
the length of

00:08:38.570 --> 00:08:42.039
that whole thing squared.

00:08:42.039 --> 00:08:45.670
And all of this is going to
be equal to H squared.

00:08:45.670 --> 00:08:48.300
Once again, just the Pythagorean
Theorem.

00:08:48.300 --> 00:08:52.820
This or this squared, which is
the height squared, is equal

00:08:52.820 --> 00:08:55.490
to your hypotenuse squared,
this is your hypotenuse

00:08:55.490 --> 00:08:57.280
squared, minus the other
side squared.

00:08:57.280 --> 00:08:58.640
This is the other
side squared.

00:08:58.640 --> 00:09:00.770
Looks a little complicated, but
it was just a projection

00:09:00.770 --> 00:09:03.800
of this guy on to that
right there.

00:09:03.800 --> 00:09:06.840
So let's see if we can simplify
this a little bit.

00:09:06.840 --> 00:09:10.410
We're just going to have to
break out some algebra or let

00:09:10.410 --> 00:09:11.390
s can do here.

00:09:11.390 --> 00:09:13.470
Well actually, not algebra,
some linear algebra.

00:09:13.470 --> 00:09:15.060
So what is this guy?

00:09:15.060 --> 00:09:18.930
Well this guy is just the dot
product of this with itself.

00:09:18.930 --> 00:09:23.610
So this is just equal to-- we
know, I mean any vector, if

00:09:23.610 --> 00:09:25.830
you take the square of its
length, it's just that vector

00:09:25.830 --> 00:09:26.660
dotted with itself.

00:09:26.660 --> 00:09:29.290
So this thing, if we are taking
the square of this

00:09:29.290 --> 00:09:30.550
guy's length, it's just
equal to this

00:09:30.550 --> 00:09:32.570
guy dotted with himself.

00:09:32.570 --> 00:09:34.950
Let me write everything
over again.

00:09:34.950 --> 00:09:45.210
So we get H squared is equal to
v2 dot v2, and then minus

00:09:45.210 --> 00:09:46.900
this guy dotted with himself.

00:09:46.900 --> 00:09:58.790
So minus v2 dot v1 over v1 dot
v1, times the vector v1,

00:09:58.790 --> 00:10:03.000
dotted with itself.

00:10:03.000 --> 00:10:07.420
Dotted with v2 dot v1--
remember, this green part is

00:10:07.420 --> 00:10:15.000
just a number-- over
v1 dot v1 times v1.

00:10:15.000 --> 00:10:16.063
And what is this equal to?

00:10:16.063 --> 00:10:17.810
Let's just simplify this.

00:10:17.810 --> 00:10:20.370
These are just scalar
quantities, and we saw that

00:10:20.370 --> 00:10:23.110
the dot product is associative
with respect to scalar

00:10:23.110 --> 00:10:25.190
quantities, so we can just
change the order here.

00:10:25.190 --> 00:10:26.640
So this is going to be
equal to the scalar

00:10:26.640 --> 00:10:28.750
quantity times itself.

00:10:28.750 --> 00:10:36.110
So we could say this is
equal to v2 dot v1.

00:10:36.110 --> 00:10:37.260
Let me write it this way.

00:10:37.260 --> 00:10:40.000
V2 dot v1, that's going to
be-- and we're going to

00:10:40.000 --> 00:10:44.660
multiply the numerator times
itself, v2 dot v1.

00:10:44.660 --> 00:10:53.240
And then all of that over v1
dot v1 times v1 dot v1.

00:10:53.240 --> 00:10:54.690
Remember, I'm just taking
these two terms and

00:10:54.690 --> 00:10:56.290
multiplying them
by each other.

00:10:56.290 --> 00:10:59.460
I'm just switching the order,
and then we know that the

00:10:59.460 --> 00:11:02.690
scalars can be taken out,
times these two

00:11:02.690 --> 00:11:03.750
guys dot each other.

00:11:03.750 --> 00:11:07.470
Times v1 dot v1.

00:11:07.470 --> 00:11:10.000
That's what this
simplifies to.

00:11:10.000 --> 00:11:11.830
Now this is now a number.

00:11:11.830 --> 00:11:14.000
We had vectors here, but when
you take a dot product, you

00:11:14.000 --> 00:11:14.650
just get a number.

00:11:14.650 --> 00:11:17.180
And this number is the
same as this number.

00:11:17.180 --> 00:11:18.780
So we can simplify
a little bit.

00:11:18.780 --> 00:11:24.270
So we can cross those two guys
out, and then we are left with

00:11:24.270 --> 00:11:30.850
that our height squared
is equal to this

00:11:30.850 --> 00:11:32.960
expression times itself.

00:11:32.960 --> 00:11:35.930
We have it times itself twice,
so it's equal to-- let me

00:11:35.930 --> 00:11:36.680
start over here.

00:11:36.680 --> 00:11:45.630
It's equal to v2 dot v2 minus
this guy times itself.

00:11:45.630 --> 00:11:53.790
So v2 dot v1 squared, all of
that over just one of these

00:11:53.790 --> 00:11:59.370
guys. v1 dot v1.

00:11:59.370 --> 00:12:01.650
That is what the height
squared is.

00:12:01.650 --> 00:12:03.480
Now we have the height squared,
we could take the

00:12:03.480 --> 00:12:05.670
square root if we just want
to solve for the height.

00:12:05.670 --> 00:12:10.350
But to keep our math simple, we
know that area is equal to

00:12:10.350 --> 00:12:12.080
base times height.

00:12:12.080 --> 00:12:14.660
Let's just say what the area
squared is equal to.

00:12:14.660 --> 00:12:18.410
Because then both of these
terms will get squared.

00:12:18.410 --> 00:12:22.024
So if the area is equal to base
times height-- we saw

00:12:22.024 --> 00:12:24.970
that at the beginning of the
video-- then the area squared

00:12:24.970 --> 00:12:26.780
is going to be equal to these
two guys squared.

00:12:26.780 --> 00:12:30.640
It's going to be equal to base
squared times height squared.

00:12:30.640 --> 00:12:33.440
Now what is the base squared?

00:12:33.440 --> 00:12:34.990
Let me do it like this.

00:12:34.990 --> 00:12:38.900
So the base squared-- we already
saw, the base of our

00:12:38.900 --> 00:12:49.590
parallelogram is the length
of vector v1.

00:12:49.590 --> 00:12:51.830
Now what is the base squared?

00:12:51.830 --> 00:12:54.580
The base squared is going
to be the length

00:12:54.580 --> 00:12:58.030
of vector v1 squared.

00:12:58.030 --> 00:13:03.730
Or another way of writing
that is v1 dot v1.

00:13:03.730 --> 00:13:06.300
And we already know what the
height squared is, it's this

00:13:06.300 --> 00:13:08.520
expression right there.

00:13:08.520 --> 00:13:09.780
Let me write that down.

00:13:09.780 --> 00:13:12.550
The height squared is the height
squared right there.

00:13:12.550 --> 00:13:15.890
So what is our area squared
going to be?

00:13:15.890 --> 00:13:21.290
Our area squared-- let me go
down here where I'll have more

00:13:21.290 --> 00:13:25.610
space-- our area squared is
going to be equal to our base

00:13:25.610 --> 00:13:30.865
squared, which is v1 dot v1
times our height squared.

00:13:33.750 --> 00:13:35.275
Let with me write
it like this.

00:13:38.780 --> 00:13:41.170
No, I was using the
wrong color.

00:13:41.170 --> 00:13:52.900
Times this guy over here. v2 dot
v2 minus v2 dot v1 squared

00:13:52.900 --> 00:13:57.830
over v1 dot v1.

00:13:57.830 --> 00:14:00.230
Now what does this
simplified to?

00:14:00.230 --> 00:14:02.670
Well, this is just a number,
these are all just numbers.

00:14:02.670 --> 00:14:05.310
So if I multiply, if I
distribute this out, this is

00:14:05.310 --> 00:14:06.590
equal to what?

00:14:06.590 --> 00:14:12.340
This times this is equal to v1--
let me color code it-- v1

00:14:12.340 --> 00:14:22.100
dot v1 times this guy
times v2 dot v2.

00:14:22.100 --> 00:14:23.680
And then when I multiplied
this guy times

00:14:23.680 --> 00:14:25.010
that guy, what happens?

00:14:25.010 --> 00:14:26.660
Well I have this guy in the
numerator and that guy in the

00:14:26.660 --> 00:14:28.010
denominator, so they
cancel out.

00:14:28.010 --> 00:14:37.700
So I'm just left with minus
v2 dot v1 squared.

00:14:37.700 --> 00:14:40.650
Now let's remind ourselves what
these two vectors were.

00:14:40.650 --> 00:14:44.650
v1 was the vector ac and
v2 is the vector bd.

00:14:44.650 --> 00:14:48.110
Let me rewrite it down here so
we have it to work with.

00:14:48.110 --> 00:14:54.760
So v1 was equal to the vector
ac, and v2 is equal

00:14:54.760 --> 00:14:57.600
to the vector bd.

00:14:57.600 --> 00:14:59.790
So what is v1 dot v1?

00:15:03.320 --> 00:15:08.750
That is equal to a dot
a, a times a, a

00:15:08.750 --> 00:15:11.020
squared plus c squared.

00:15:11.020 --> 00:15:12.990
That's this, right there.

00:15:12.990 --> 00:15:14.800
And that's what? v2 dot v2.

00:15:14.800 --> 00:15:19.740
So we're going to have
that times v2 dot v2.

00:15:19.740 --> 00:15:25.030
v2 dot v2 is v squared
plus d squared.

00:15:25.030 --> 00:15:30.450
And then we're going to have
minus v2 dot v1 squared.

00:15:30.450 --> 00:15:34.080
So what's v2 dot v1?

00:15:34.080 --> 00:15:38.590
It's b times a, plus d times c,
or a times b plus -- we're

00:15:38.590 --> 00:15:39.850
just dotting these two guys.

00:15:39.850 --> 00:15:46.600
So it's ab plus cd, and then
we're squaring it.

00:15:46.600 --> 00:15:48.910
And let's see what this
simplifies to.

00:15:48.910 --> 00:15:51.630
Hopefully it simplifies
to something.

00:15:51.630 --> 00:15:53.050
Let me switch colors.

00:15:53.050 --> 00:15:55.550
So if we just multiply this
out, let me write it here.

00:15:55.550 --> 00:15:59.185
Our area squared is equal to
a squared times b squared.

00:16:01.930 --> 00:16:08.150
a squared times d squared,
plus c squared times b

00:16:08.150 --> 00:16:13.670
squared, plus c squared
times d squared.

00:16:17.350 --> 00:16:21.220
And then minus this
guy squared.

00:16:21.220 --> 00:16:22.960
What is that going
to be equal to?

00:16:22.960 --> 00:16:26.620
ab squared is a squared,
b squared.

00:16:26.620 --> 00:16:29.650
And then I'm going to multiply
these guys times each other

00:16:29.650 --> 00:16:35.750
twice, so that's going
to be plus 2abcd.

00:16:35.750 --> 00:16:40.020
And then you're going to have
a plus c squared, d squared.

00:16:40.020 --> 00:16:42.260
I just foiled this out, that's
the best way you

00:16:42.260 --> 00:16:44.010
could think about it.

00:16:44.010 --> 00:16:47.920
And then, if I distribute this
negative sign, what do I have?

00:16:47.920 --> 00:16:49.230
Let me rewrite everything.

00:16:49.230 --> 00:16:53.380
It's equal to a squared b
squared, plus a squared d

00:16:53.380 --> 00:16:59.100
squared, plus c squared b
squared, plus c squared d

00:16:59.100 --> 00:17:08.060
squared, minus a squared b
squared, minus 2abcd, minus c

00:17:08.060 --> 00:17:09.200
squared, d squared.

00:17:09.200 --> 00:17:11.920
All I did is, I distributed
the minus sign.

00:17:11.920 --> 00:17:14.960
Now it looks like some things
will simplify nicely.

00:17:14.960 --> 00:17:17.550
And now remember, all this is
equal to our area squared.

00:17:17.550 --> 00:17:21.180
We have a ab squared, we have
a minus ab squared.

00:17:21.180 --> 00:17:22.589
They cancel out.

00:17:22.589 --> 00:17:27.130
We have a minus cd squared
and a cd squared,

00:17:27.130 --> 00:17:28.650
so they cancel out.

00:17:28.650 --> 00:17:32.100
So all we're left with is that
the area of our parallelogram

00:17:32.100 --> 00:17:45.490
squared is equal to a squared
d squared minus 2abcd plus c

00:17:45.490 --> 00:17:47.740
squared b squared.

00:17:51.920 --> 00:17:55.910
Now this might look a little bit
bizarre to you, but if you

00:17:55.910 --> 00:17:59.210
made a substitution right here,
if you said that x is

00:17:59.210 --> 00:18:04.540
equal to ad, and if you said y
is equal to cb, then what does

00:18:04.540 --> 00:18:06.730
this become?

00:18:06.730 --> 00:18:10.220
This is equal to x
squared minus 2

00:18:10.220 --> 00:18:14.180
times xy plus y squared.

00:18:14.180 --> 00:18:14.750
Right?

00:18:14.750 --> 00:18:15.740
Hopefully you recognize this.

00:18:15.740 --> 00:18:20.610
And this is just the same thing
as x minus y squared.

00:18:20.610 --> 00:18:23.580
So, if this is our substitutions
we made-- I did

00:18:23.580 --> 00:18:26.940
this just so you can visualize
this a little bit better.

00:18:26.940 --> 00:18:32.980
So we have our area squared is
equal to x minus y squared or

00:18:32.980 --> 00:18:39.640
ad minus cb, or let me
write it, bc squared.

00:18:39.640 --> 00:18:43.960
That's what the area of our
parallelogram squared is.

00:18:43.960 --> 00:18:46.540
And if you don't quite
understand what I did here, I

00:18:46.540 --> 00:18:47.890
just made these substitutions
so you can

00:18:47.890 --> 00:18:49.080
recognize it better.

00:18:49.080 --> 00:18:51.010
But just understand that this
is the same thing as this.

00:18:51.010 --> 00:18:54.100
If you want, you can just
multiply this guy out and

00:18:54.100 --> 00:18:55.670
you'll get that right there.

00:18:55.670 --> 00:18:58.650
But what is this?

00:18:58.650 --> 00:19:00.700
What is this thing right here?

00:19:00.700 --> 00:19:02.250
It's the determinant.

00:19:02.250 --> 00:19:05.470
This is the determinant of
our original matrix.

00:19:09.380 --> 00:19:12.710
Well, I called that matrix A
and then I used A again for

00:19:12.710 --> 00:19:15.460
area, so let me write
it this way.

00:19:15.460 --> 00:19:19.880
Area squared -- let me
write it like this.

00:19:19.880 --> 00:19:27.870
Area squared is equal to
ad minus bc squared.

00:19:27.870 --> 00:19:31.940
So this is area, these
A's are all area.

00:19:31.940 --> 00:19:32.940
But what is this?

00:19:32.940 --> 00:19:34.770
This is the determinant
of my matrix.

00:19:34.770 --> 00:19:43.050
That is the determinant of my
matrix A, my original matrix

00:19:43.050 --> 00:19:45.680
that I started the problem with,
which is equal to the

00:19:45.680 --> 00:19:48.380
determinant of abcd.

00:19:48.380 --> 00:19:48.870
Right?

00:19:48.870 --> 00:19:54.340
The determinant of this is ad
minus bc, by definition.

00:19:54.340 --> 00:19:59.190
So your area-- this
is exciting!

00:19:59.190 --> 00:20:02.140
So the area of your
parallelogram squared is equal

00:20:02.140 --> 00:20:06.070
to the determinant of the matrix
whose column vectors

00:20:06.070 --> 00:20:08.210
construct that parallelogram.

00:20:08.210 --> 00:20:12.650
Is equal to the determinant
of your matrix squared.

00:20:12.650 --> 00:20:14.690
Or if you take the square root
of both sides, you get the

00:20:14.690 --> 00:20:19.980
area is equal to the absolute
value of the determinant of A.

00:20:19.980 --> 00:20:22.030
Which is a pretty neat
outcome, especially

00:20:22.030 --> 00:20:24.732
considering how much hairy
algebra we had to go through.

00:20:24.732 --> 00:20:28.720
Let's go back all the way over
here, go back to the drawing.

00:20:28.720 --> 00:20:34.740
So if we want to figure out the
area of this parallelogram

00:20:34.740 --> 00:20:39.610
right here, that is defined, or
that is created, by the two

00:20:39.610 --> 00:20:44.620
column vectors of a matrix, we
literally just have to find

00:20:44.620 --> 00:20:46.460
the determinant of the matrix.

00:20:46.460 --> 00:20:51.620
The area of this is equal to
the absolute value of the

00:20:51.620 --> 00:20:52.630
determinant of A.

00:20:52.630 --> 00:20:54.060
And you have to do that because

00:20:54.060 --> 00:20:55.930
this might be negative.

00:20:55.930 --> 00:20:58.830
You can imagine if you swapped
these guys around, if you

00:20:58.830 --> 00:21:01.150
swapped some of the rows, this
guy would be negative, but you

00:21:01.150 --> 00:21:02.230
can 't have a negative area.

00:21:02.230 --> 00:21:04.560
And it wouldn't really change
the definition, it really

00:21:04.560 --> 00:21:05.730
wouldn't change what spanned.

00:21:05.730 --> 00:21:08.960
If you switched v1 and v2,
you're still spanning the same

00:21:08.960 --> 00:21:11.470
parallelogram, you just might
get the negative of the

00:21:11.470 --> 00:21:12.770
determinant.

00:21:12.770 --> 00:21:14.640
But that is a really
neat outcome.

00:21:14.640 --> 00:21:17.240
And you know, when you first
learned determinants in

00:21:17.240 --> 00:21:20.000
school-- I mean, we learned
the first motivation for a

00:21:20.000 --> 00:21:24.640
determinant was this idea of
when we take the inverse of a

00:21:24.640 --> 00:21:27.110
2 by 2, this thing shows up in
the denominator and we call

00:21:27.110 --> 00:21:28.180
that the determinant.

00:21:28.180 --> 00:21:30.470
But now there's this other
interpretation here.

00:21:30.470 --> 00:21:33.590
There's actually the area of the
parallelogram created by

00:21:33.590 --> 00:21:36.360
the column vectors
of this matrix.


Simpler 4x4 determinant

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:03.670
I have this 4 by 4
matrix, A, here.

00:00:03.670 --> 00:00:07.120
And let's see if we can figure
out its determinant, the

00:00:07.120 --> 00:00:08.530
determinant of A.

00:00:08.530 --> 00:00:11.370
And before just doing it the way
we've done it in the past,

00:00:11.370 --> 00:00:13.530
where you go down one of the
rows or one of the columns--

00:00:13.530 --> 00:00:16.890
and you notice, there's no 0's
here, so there's no easy row

00:00:16.890 --> 00:00:19.250
or easy column to take
the determinant by.

00:00:19.250 --> 00:00:21.430
We could have gone down this
row and do all the

00:00:21.430 --> 00:00:23.510
submatrices, but this becomes
pretty involved.

00:00:23.510 --> 00:00:27.740
You end up doing four 3 by 3
determinants, and then each of

00:00:27.740 --> 00:00:29.880
those are composed of three
2 by 2 determinants.

00:00:29.880 --> 00:00:32.030
It becomes a pretty
hairy process.

00:00:32.030 --> 00:00:35.230
Let's see if we can use some
of the realizations we've

00:00:35.230 --> 00:00:36.720
discovered in the last
two videos to

00:00:36.720 --> 00:00:38.250
simplify this process.

00:00:38.250 --> 00:00:41.270
Well, one of the realizations is
that row operations, or if

00:00:41.270 --> 00:00:50.470
you subtract-- let me write it
this way-- if you replace row

00:00:50.470 --> 00:00:58.590
j with, let's say, row j minus
some scale or multiple, times

00:00:58.590 --> 00:01:07.200
row i, it does not change
the determinant of A.

00:01:07.200 --> 00:01:09.300
We saw that, I think it
was two videos ago.

00:01:09.300 --> 00:01:10.880
So this is a pretty
big realization.

00:01:10.880 --> 00:01:13.810
We can do these type of row
operations and it won't change

00:01:13.810 --> 00:01:14.700
the determinant.

00:01:14.700 --> 00:01:17.730
The other realization we had
was that these upper

00:01:17.730 --> 00:01:26.220
triangular matrices, you can
figure out their determinant.

00:01:26.220 --> 00:01:27.940
So what does upper triangular
look like?

00:01:27.940 --> 00:01:28.900
Let me just review it.

00:01:28.900 --> 00:01:31.370
The upper triangular--
everything below the diagonal.

00:01:31.370 --> 00:01:34.850
So let's say the diagonal has--
let me just draw its

00:01:34.850 --> 00:01:35.710
terms like that.

00:01:35.710 --> 00:01:39.380
These are some non-zero terms.
Oh, they don't have to be.

00:01:39.380 --> 00:01:43.920
Then upper triangular,
everything below the diagonal

00:01:43.920 --> 00:01:50.630
is a 0, and everything above the
diagonal probably isn't a

00:01:50.630 --> 00:01:51.740
0, but you never know.

00:01:51.740 --> 00:01:55.400
But they're non-zero terms, so
all the red stuff here is

00:01:55.400 --> 00:01:57.940
non-zero, all this stuff
in green is 0.

00:01:57.940 --> 00:02:00.270
And I didn't touch on it in that
video, but there is also

00:02:00.270 --> 00:02:05.630
such a thing as a lower
triangular, that you might

00:02:05.630 --> 00:02:06.800
have guessed how it looks.

00:02:06.800 --> 00:02:10.370
Everything above the main
diagonal is 0, so this is the

00:02:10.370 --> 00:02:13.810
main diagonal right here, all
the way down like that.

00:02:13.810 --> 00:02:17.060
All of these guys are going
to be non-zero.

00:02:17.060 --> 00:02:19.730
All of that's going to be
non-zero, and then the 0's are

00:02:19.730 --> 00:02:24.630
going to be above the
diagonal, like that.

00:02:24.630 --> 00:02:27.660
We saw in the last video that
the determinant of this guy is

00:02:27.660 --> 00:02:31.130
just equal to the product of the
diagonal entries, which is

00:02:31.130 --> 00:02:35.050
a very simple way of finding
a determinant.

00:02:35.050 --> 00:02:37.160
And you could use the same
argument we made in the last

00:02:37.160 --> 00:02:39.540
video to say that the same is
true of the lower triangular

00:02:39.540 --> 00:02:42.640
matrix, that its determinant
is also just the product of

00:02:42.640 --> 00:02:44.170
those entries.

00:02:44.170 --> 00:02:46.030
I won't prove it here, but you
can use the exact same

00:02:46.030 --> 00:02:48.980
argument you used in the video
that I just did on the upper

00:02:48.980 --> 00:02:50.300
triangular.

00:02:50.300 --> 00:02:52.880
So given this, that the
determinant of this is just

00:02:52.880 --> 00:02:55.780
the product of those guys, and
that I can perform row

00:02:55.780 --> 00:02:59.250
operations on this guy and not
change the determinant, maybe

00:02:59.250 --> 00:03:02.070
a simpler way to calculate this
determinant is to get

00:03:02.070 --> 00:03:04.860
this guy into an upper
triangular form, and then just

00:03:04.860 --> 00:03:07.360
multiply the entries
down the diagonal.

00:03:07.360 --> 00:03:09.060
So let's do that.

00:03:09.060 --> 00:03:11.085
So we want to find the
determinant of A.

00:03:15.210 --> 00:03:15.870
Let me rewrite A right here.

00:03:15.870 --> 00:03:30.740
It's 1, 2, 2, 1, 1, 1, 2, 4,
2, 2, 7, 5, 2, minus 1, 4,

00:03:30.740 --> 00:03:33.650
minus 6, 3.

00:03:33.650 --> 00:03:38.950
Now let's try to get this into
upper triangular form.

00:03:38.950 --> 00:03:43.570
So let's replace the second
row with the-- so I'm just

00:03:43.570 --> 00:03:45.630
going to keep the first
row the same.

00:03:45.630 --> 00:03:48.790
1, 2, 2, 1.

00:03:48.790 --> 00:03:52.900
And let's replace the second row
with the second row minus

00:03:52.900 --> 00:03:54.230
the first row.

00:03:54.230 --> 00:03:57.100
The second row minus the first
row is going to be equal to 1

00:03:57.100 --> 00:03:58.960
minus 1 is 0.

00:03:58.960 --> 00:04:01.610
So in this case the constant
is just 1.

00:04:01.610 --> 00:04:02.530
So 1 minus 1 is 0.

00:04:02.530 --> 00:04:05.120
2 minus 2 is 0.

00:04:05.120 --> 00:04:07.430
4 minus 2 is 2.

00:04:07.430 --> 00:04:09.620
2 minus 1 is 1.

00:04:09.620 --> 00:04:15.040
Now let's replace the third row
with the third row minus 2

00:04:15.040 --> 00:04:17.560
times the second row.

00:04:17.560 --> 00:04:21.390
So 2 minus 2 times 1 is 0.

00:04:21.390 --> 00:04:25.100
7 minus 2 times 2 is 3.

00:04:25.100 --> 00:04:29.590
5 minus 2 times 2 is 1.

00:04:29.590 --> 00:04:36.160
2 minus 2 times 1 is 0.

00:04:36.160 --> 00:04:37.845
Let me get a good color here.

00:04:37.845 --> 00:04:39.540
I'll do pink.

00:04:39.540 --> 00:04:44.630
Let's replace the last row
with the last row,

00:04:44.630 --> 00:04:46.870
essentially, plus
the first row.

00:04:46.870 --> 00:04:49.410
You could say minus minus 1
times the first row is the

00:04:49.410 --> 00:04:52.000
same thing as the last row
plus the first row.

00:04:52.000 --> 00:04:55.680
So minus 1 plus 1 is 0.

00:04:55.680 --> 00:04:58.700
4 plus 2 is 6.

00:04:58.700 --> 00:05:04.960
Minus 6 plus 2 is minus 4.

00:05:04.960 --> 00:05:09.490
And then 3 plus 1 is 4.

00:05:09.490 --> 00:05:11.220
So there we have it like that.

00:05:11.220 --> 00:05:13.810
And this guy has two 0's
here, so maybe I want

00:05:13.810 --> 00:05:15.530
to swap some rows.

00:05:15.530 --> 00:05:16.920
So let me swap some rows.

00:05:16.920 --> 00:05:20.200
So if we swap rows,
what happens?

00:05:20.200 --> 00:05:22.650
I'm going to swap the middle
two rows just for fun.

00:05:22.650 --> 00:05:24.090
Well, not just for fun.

00:05:24.090 --> 00:05:26.560
Because I want a pivot
entry right here.

00:05:26.560 --> 00:05:27.520
I shouldn't say a pivot entry.

00:05:27.520 --> 00:05:29.105
I want to do it in upper
triangular form.

00:05:29.105 --> 00:05:30.965
So I want a non-zero
entry here.

00:05:30.965 --> 00:05:33.370
This is a 0, so I'm going
to move this guy down.

00:05:33.370 --> 00:05:34.770
So I'm going to keep the
top row the same.

00:05:34.770 --> 00:05:36.680
1, 2, 2, 1.

00:05:36.680 --> 00:05:38.190
I'm going to keep the
bottom row the same.

00:05:38.190 --> 00:05:43.205
0, 0, 6, minus 4, 4.

00:05:43.205 --> 00:05:46.630
And I'm going to swap these
guys right here.

00:05:46.630 --> 00:05:51.150
So this is going to
be 0, 3, 1, 0.

00:05:51.150 --> 00:05:54.630
And then 0, 0, 2, 1.

00:05:54.630 --> 00:05:57.130
Now, can I just swap
entries like that?

00:05:57.130 --> 00:05:59.520
Well, I can, but you have to
remember that when you swap

00:05:59.520 --> 00:06:02.130
entries, your resulting
determinant is going to be the

00:06:02.130 --> 00:06:03.810
negative of your original
determinant.

00:06:03.810 --> 00:06:06.990
So if we swap these two guys,
the determinant of this is

00:06:06.990 --> 00:06:10.200
going to be the negative
of this determinant.

00:06:10.200 --> 00:06:13.110
When you swap two rows, you
just flip the sign of the

00:06:13.110 --> 00:06:14.100
determinant.

00:06:14.100 --> 00:06:14.660
We saw that.

00:06:14.660 --> 00:06:16.370
That was one of the first videos
we did on these, kind

00:06:16.370 --> 00:06:18.410
of messing with the
determinants.

00:06:18.410 --> 00:06:20.200
Now, what do we want
to do here?

00:06:20.200 --> 00:06:26.530
To get this guy into upper
triangular form, it'd be nice

00:06:26.530 --> 00:06:28.820
to get this to be a 0.

00:06:28.820 --> 00:06:31.520
So to get that to be
a 0, let me keep

00:06:31.520 --> 00:06:32.960
everything else the same.

00:06:32.960 --> 00:06:35.840
So I have a 1, 2, 2, 1.

00:06:35.840 --> 00:06:39.240
I have a 0, 3, 1, 0.

00:06:39.240 --> 00:06:43.680
The third row is 0, 0, 2, 1.

00:06:43.680 --> 00:06:47.470
And now this last row, let me
replace it with the last row

00:06:47.470 --> 00:06:49.270
minus 3 times this row.

00:06:49.270 --> 00:06:51.150
So let me write it like this.

00:06:51.150 --> 00:06:53.300
I have to carry that negative
sign as well.

00:06:53.300 --> 00:06:56.700
So I'm going to replace this
last row with the last row

00:06:56.700 --> 00:07:02.750
minus 2 times the second row.

00:07:02.750 --> 00:07:04.610
want to zero it out.

00:07:04.610 --> 00:07:07.830
So 0 minus 2 times 0 is 0.

00:07:07.830 --> 00:07:10.680
6 minus 2 times 3 is 0.

00:07:10.680 --> 00:07:15.280
Minus 4 minus 2 times
1 is minus 6.

00:07:15.280 --> 00:07:18.590
And then 4 minus 2 times
0 is just 4.

00:07:18.590 --> 00:07:20.460
We're almost there.

00:07:20.460 --> 00:07:25.040
Now we want to zero this
guy out, so let's

00:07:25.040 --> 00:07:27.140
replace this one.

00:07:27.140 --> 00:07:30.560
So I'm going to keep my top
three rows the same again.

00:07:30.560 --> 00:07:32.150
And let me see if I can write
it a little bit neater.

00:07:32.150 --> 00:07:36.800
So my first row is 1, 2, 2, 1.

00:07:36.800 --> 00:07:41.010
My second row is 0, 3, 1, 0.

00:07:41.010 --> 00:07:44.930
Fourth row is 0, 0, 2, 1.

00:07:44.930 --> 00:07:46.540
And I'm going to take
the matrix.

00:07:46.540 --> 00:07:47.960
I haven't written the
fourth row yet.

00:07:47.960 --> 00:07:51.250
And, of course, the negative
of this is going to be

00:07:51.250 --> 00:07:53.540
determinant of our original
matrix, because we had swapped

00:07:53.540 --> 00:07:54.650
those rows.

00:07:54.650 --> 00:08:00.790
But let's replace this last row
with the last row, plus 3

00:08:00.790 --> 00:08:03.650
times the third row.

00:08:03.650 --> 00:08:06.240
So we get 0 plus
3 times 0 is 0.

00:08:06.240 --> 00:08:08.710
0 plus 3 times 0 is 0.

00:08:08.710 --> 00:08:12.220
Minus 6 plus 3 times 2 is 0.

00:08:12.220 --> 00:08:15.740
4 plus 3 times 1 is 7.

00:08:15.740 --> 00:08:18.620
And just like that, we have a
determinant of a matrix in

00:08:18.620 --> 00:08:20.600
upper triangular form.

00:08:20.600 --> 00:08:28.080
So this is going to be equal to
the product of these guys.

00:08:28.080 --> 00:08:29.945
We can't forget our
negative sign.

00:08:29.945 --> 00:08:32.950
Let's throw our negative sign
out there and put a

00:08:32.950 --> 00:08:34.500
parentheses just like that.

00:08:34.500 --> 00:08:36.039
This is going to be
the product of

00:08:36.039 --> 00:08:37.030
that diagonal entry.

00:08:37.030 --> 00:08:43.760
1 times 3, times 3, times 2,
times 7, which is 6 times 7,

00:08:43.760 --> 00:08:46.130
which is 42.

00:08:46.130 --> 00:08:50.710
So the determinant of this
matrix is minus 42, which was

00:08:50.710 --> 00:08:53.100
pretty fast. This was a
pretty fast shortcut.

00:08:53.100 --> 00:08:56.080
And it actually turns out it
tends to be computationally

00:08:56.080 --> 00:08:59.580
more efficient to use these
takeaways to put things into

00:08:59.580 --> 00:09:02.740
upper triangular form first. And
then, you know, if you do

00:09:02.740 --> 00:09:05.300
swaps, you have to remember to
make the determinant negative.

00:09:05.300 --> 00:09:07.910
And then just multiply
down the diagonal.

00:09:07.910 --> 00:09:09.650
And we did that there, and
we got the determinant

00:09:09.650 --> 00:09:12.250
as being minus 42.


Upper triangular determinant

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:04.810
Let's say I have a matrix where
everything below the

00:00:04.810 --> 00:00:06.530
main diagonal is a 0.

00:00:06.530 --> 00:00:08.610
And I'll start-- just for the
sake of argument, let's start

00:00:08.610 --> 00:00:09.860
with a 2 by 2 matrix.

00:00:12.460 --> 00:00:17.930
I have the values
a, b, 0, and d.

00:00:17.930 --> 00:00:21.180
Instead of a c, I have a 0
there, so everything below the

00:00:21.180 --> 00:00:23.260
main diagonal is a 0.

00:00:23.260 --> 00:00:25.190
What is the determinant
of this going to be?

00:00:25.190 --> 00:00:27.590
Let's call that matrix a.

00:00:27.590 --> 00:00:34.000
The determinant of a is going
to be equal to ad

00:00:34.000 --> 00:00:35.890
minus b times 0.

00:00:35.890 --> 00:00:37.900
That's just a 0, so you don't
have to write it.

00:00:37.900 --> 00:00:40.030
It's equal to a times d.

00:00:40.030 --> 00:00:41.710
Now, let's say I have
another matrix.

00:00:41.710 --> 00:00:43.332
Let's call it b.

00:00:43.332 --> 00:00:45.855
We'll say it's a
3 by 3 matrix.

00:00:48.950 --> 00:00:53.860
And let's say its entries
are a, b, c.

00:00:53.860 --> 00:00:56.750
We've got a 0 here.

00:00:56.750 --> 00:01:00.360
Then you-- let's say you have
a d here, e, then you have

00:01:00.360 --> 00:01:04.660
another 0 here, another 0
here, and you have an f.

00:01:04.660 --> 00:01:09.510
Once again, all of the entries
below the main diagonal are 0.

00:01:09.510 --> 00:01:11.420
What's this guy's determinant?

00:01:11.420 --> 00:01:13.550
Well, we learned several videos
ago that you can always

00:01:13.550 --> 00:01:16.160
pick the row in the column that
has the most 0's on it.

00:01:16.160 --> 00:01:18.240
That simplifies your
situation.

00:01:18.240 --> 00:01:22.330
Let's find the determinant along
this column right here.

00:01:22.330 --> 00:01:28.660
The determinant of b is going
to be equal to a times the

00:01:28.660 --> 00:01:33.030
submatrix if you were to ignore
a's row and column.

00:01:33.030 --> 00:01:42.410
a times the determinant of d,
e, 0, f, and then minus 0

00:01:42.410 --> 00:01:44.550
times its submatrix.

00:01:44.550 --> 00:01:46.570
You could cancel out-- or times
the determinant of its

00:01:46.570 --> 00:01:48.830
submatrix, that row
and that column.

00:01:48.830 --> 00:01:50.190
You'd get b, c, 0, f.

00:01:52.850 --> 00:01:57.980
And then you have plus 0 times--
you get rid of that

00:01:57.980 --> 00:02:00.120
row, that column, you
get b, c, d, e.

00:02:02.870 --> 00:02:04.590
Obviously, these two guys
are going to be 0.

00:02:04.590 --> 00:02:07.760
I don't care what these 2 by
2 matrices-- what their

00:02:07.760 --> 00:02:09.539
determinants end up
evaluating to.

00:02:09.539 --> 00:02:11.850
These are both going to be
equal to 0, because we're

00:02:11.850 --> 00:02:13.380
multiplying by 0.

00:02:13.380 --> 00:02:16.100
We're left with a times the
determinant of this, and the

00:02:16.100 --> 00:02:18.670
determinant of this is pretty
straightforward.

00:02:18.670 --> 00:02:20.670
We're going to have-- it's just
going to be equal to a

00:02:20.670 --> 00:02:24.550
times the determinant of this,
which is df minus 0 times e.

00:02:24.550 --> 00:02:25.800
It's just going to be df.

00:02:27.770 --> 00:02:31.240
The determinant of b is adf.

00:02:31.240 --> 00:02:34.640
Notice that the determinant
of a was just a and d.

00:02:34.640 --> 00:02:36.430
Now, you might see a pattern.

00:02:36.430 --> 00:02:40.590
In both cases we had 0's below
the main diagonal, right?

00:02:40.590 --> 00:02:43.260
This was the main diagonal
right here.

00:02:43.260 --> 00:02:45.250
And when we took the
determinants of the matrix,

00:02:45.250 --> 00:02:52.140
the determinant just ended up
being the product of the

00:02:52.140 --> 00:02:55.230
entries along the
main diagonal.

00:02:55.230 --> 00:02:57.390
And if you think that that's
a general trend that always

00:02:57.390 --> 00:02:59.290
applies, you are correct.

00:02:59.290 --> 00:03:02.130
We can do it in the
general case.

00:03:02.130 --> 00:03:04.340
Let's do it with our
general case.

00:03:04.340 --> 00:03:11.710
Let's say we have some matrix,
a, and it is equal to a, 1, 1.

00:03:11.710 --> 00:03:14.460
And you have a, 2, 2.

00:03:14.460 --> 00:03:16.390
You're going to have
a 0 right there.

00:03:16.390 --> 00:03:19.660
And then you just keep going all
the way down to a, n, n.

00:03:19.660 --> 00:03:23.560
In this row, everything's going
to be a 0, except for

00:03:23.560 --> 00:03:24.920
that last column.

00:03:24.920 --> 00:03:26.960
This is all a 0 right here.

00:03:26.960 --> 00:03:32.120
So everything below the main
diagonal is a 0, just like

00:03:32.120 --> 00:03:35.720
this one, but we're doing it
in the general n by n case.

00:03:35.720 --> 00:03:40.075
And everything up here is--
well, it doesn't have to be 0.

00:03:40.075 --> 00:03:43.980
This is a, 1, 2, all
the way to a, 1, n.

00:03:43.980 --> 00:03:46.530
This is a, 2, n.

00:03:46.530 --> 00:03:47.320
Keep going down.

00:03:47.320 --> 00:03:51.070
So everything at the main
diagonal or above isn't

00:03:51.070 --> 00:03:52.680
necessarily equal to 0.

00:03:52.680 --> 00:03:54.440
If you wanted to find the
determinant of a, we could do

00:03:54.440 --> 00:03:56.410
the same thing we did here.

00:03:56.410 --> 00:04:02.270
We could go down that first
row right there.

00:04:02.270 --> 00:04:08.200
The determinant of our matrix,
a, is equal to this guy-- a,

00:04:08.200 --> 00:04:11.960
1, 1-- times the determinant
of its submatrix.

00:04:11.960 --> 00:04:14.895
That's going to be a, 2, 2.

00:04:14.895 --> 00:04:20.329
It goes all the way to a, 2, n,
and then a, 3, 3, all the

00:04:20.329 --> 00:04:24.410
way to a, n, n.

00:04:24.410 --> 00:04:29.190
And then, everything down here
is-- these are all 0's.

00:04:29.190 --> 00:04:32.160
Once again, we have another
situation where all of the

00:04:32.160 --> 00:04:35.940
entries below the main
diagonal are 0.

00:04:35.940 --> 00:04:38.480
What's the determinant of
this guy right here?

00:04:38.480 --> 00:04:39.860
And what-- you might say
hey, what about the

00:04:39.860 --> 00:04:40.570
rest of that row?

00:04:40.570 --> 00:04:42.850
Well, the rest of the row is
just a bunch of 0's, just like

00:04:42.850 --> 00:04:43.570
we had here.

00:04:43.570 --> 00:04:47.435
0 times the determinant of its
submatrix, and then that would

00:04:47.435 --> 00:04:48.640
be a minus and a plus.

00:04:48.640 --> 00:04:51.260
0 times the determinant of its
submatrix, so on and so forth.

00:04:51.260 --> 00:04:54.900
We just have to pay attention
to this term right there.

00:04:54.900 --> 00:04:57.360
Now, the same argument
we can do here.

00:04:57.360 --> 00:05:02.090
To find this determinant, we
can just go down that row.

00:05:02.090 --> 00:05:04.870
The determinant of this is just
going to be equal to--

00:05:04.870 --> 00:05:09.010
let's write out-- let's not
forget our a, 1, 1 out there.

00:05:09.010 --> 00:05:14.187
The determinant of this is going
to be a, 2, 2 times the

00:05:14.187 --> 00:05:15.370
determinant of its submatrix.

00:05:15.370 --> 00:05:17.290
Get rid of its row and its
column, and you're just left

00:05:17.290 --> 00:05:21.880
with a, 3, 3 all the way
down to a, n, n.

00:05:21.880 --> 00:05:25.520
Everything up here is non-zero,
so its a, 3n.

00:05:25.520 --> 00:05:28.440
Then everything below the
diagonal, once again, is just

00:05:28.440 --> 00:05:29.960
a bunch of 0's.

00:05:29.960 --> 00:05:31.630
Everything down here
is a bunch of 0's.

00:05:31.630 --> 00:05:34.650
This is another of
what we call an

00:05:34.650 --> 00:05:36.090
upper triangular matrix.

00:05:36.090 --> 00:05:37.110
Let me write that down.

00:05:37.110 --> 00:05:39.980
This whole class, where you
have 0's below the main

00:05:39.980 --> 00:05:45.200
diagonal, these are called upper
triangular matrices.

00:05:50.710 --> 00:05:53.080
Matrices, just like that.

00:05:53.080 --> 00:05:55.940
Now, we keep doing the process
over and over again.

00:05:55.940 --> 00:05:58.100
If you just keep following this
pattern over and again,

00:05:58.100 --> 00:06:01.250
now you're going to have the
determinant of this is a, 3, 3

00:06:01.250 --> 00:06:02.400
times its submatrix.

00:06:02.400 --> 00:06:03.820
And every time, the
submatrix is

00:06:03.820 --> 00:06:05.440
getting smaller and smaller.

00:06:05.440 --> 00:06:12.160
You'll eventually get to a, 1,
1 times a, 2, 2 times-- all

00:06:12.160 --> 00:06:22.950
the way to an minus 2 times
a 2 by 2 matrix over here.

00:06:22.950 --> 00:06:29.320
This is going to be an minus
1, n minus 1, an.

00:06:29.320 --> 00:06:33.260
Then this is going to be
a sub n minus 1, n.

00:06:33.260 --> 00:06:34.770
Then you're going to have
a 0 right here.

00:06:34.770 --> 00:06:37.130
It's just the bottom right-hand
corner of our

00:06:37.130 --> 00:06:39.750
original matrix, is what you're
going to be left with.

00:06:39.750 --> 00:06:41.890
And what is the determinant
of this?

00:06:41.890 --> 00:06:43.900
Well, it's just the product
of these two things.

00:06:43.900 --> 00:06:46.820
It's just this guy times this
guy minus this guy times that

00:06:46.820 --> 00:06:49.480
guy, but that's just 0.

00:06:49.480 --> 00:06:56.080
The determinant of a ends up
becoming a, 1, 1 times a, 2,

00:06:56.080 --> 00:07:01.520
2, all the way to a, n, n, or
the product of all of the

00:07:01.520 --> 00:07:04.430
entries of the main diagonal.

00:07:04.430 --> 00:07:06.850
Which is a super important take
away, because it really

00:07:06.850 --> 00:07:09.210
simplifies finding the
determinants of what would

00:07:09.210 --> 00:07:11.880
otherwise be really hard
matrices to find the

00:07:11.880 --> 00:07:12.300
determinants of.

00:07:12.300 --> 00:07:14.830
You could imagine if this
was a 100 by 100 matrix.

00:07:14.830 --> 00:07:17.300
Now, we could just multiply
the diagonal.

00:07:17.300 --> 00:07:20.020
Just to make sure that
things are clear,

00:07:20.020 --> 00:07:21.725
let me do an example.

00:07:21.725 --> 00:07:27.230
Let's say we find the
determinant of 7, 3, 4, 2.

00:07:27.230 --> 00:07:29.340
So we have 0's here.

00:07:29.340 --> 00:07:34.650
This is a minus 2, 1, and a 3,
a 0 here-- sorry, we don't

00:07:34.650 --> 00:07:35.750
want 0's there.

00:07:35.750 --> 00:07:37.190
We don't need to
have 0's there.

00:07:37.190 --> 00:07:40.480
6, 7-- we actually could have
0's there, but we don't need

00:07:40.480 --> 00:07:41.950
to have 0's there.

00:07:41.950 --> 00:07:44.780
And a 0 there, and we
have 0's there.

00:07:44.780 --> 00:07:45.500
Just like that.

00:07:45.500 --> 00:07:45.650
So

00:07:45.650 --> 00:07:48.680
its upper triangular matrix--
if you want to evaluate this

00:07:48.680 --> 00:07:53.220
determinant, you just multiply
these entries right here.

00:07:53.220 --> 00:07:58.880
The determinant is equal to 7
times minus 2 times 1 times 3.

00:07:58.880 --> 00:08:04.730
So it's 7 times minus 6 which
is equal to minus 42.

00:08:04.730 --> 00:08:06.740
And it's that easy.


Determinant after row operations

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:02.580
I have a matrix A.

00:00:02.580 --> 00:00:05.430
It is an n by n matrix.

00:00:05.430 --> 00:00:07.620
And let me just write
its rows like this.

00:00:07.620 --> 00:00:10.070
Let me just write it as r1.

00:00:10.070 --> 00:00:13.210
We could call them row vectors
maybe. r2, I'm not doing it

00:00:13.210 --> 00:00:13.790
too formally.

00:00:13.790 --> 00:00:15.510
This is just to save
on writing.

00:00:15.510 --> 00:00:19.806
And then it has an ith
row, ri, and then you

00:00:19.806 --> 00:00:21.030
can keep going .

00:00:21.030 --> 00:00:21.940
That's an i right there.

00:00:21.940 --> 00:00:26.120
Then it has a jth row, rj, and
you keep going and you get to

00:00:26.120 --> 00:00:26.780
the nth throw.

00:00:26.780 --> 00:00:28.840
It has n rows and n columns.

00:00:28.840 --> 00:00:31.330
So you get to rn
just like that.

00:00:31.330 --> 00:00:34.130
That is my matrix.

00:00:34.130 --> 00:00:40.450
Just to make sure you get what
I'm saying, so if I have a kth

00:00:40.450 --> 00:00:43.280
r-sub-k is equal to ak1.

00:00:46.510 --> 00:00:48.050
Maybe I'll write
it as a vector.

00:00:48.050 --> 00:00:53.380
ak2 all the way to akn.

00:00:53.380 --> 00:00:55.150
So this is just your standard
representation.

00:00:55.150 --> 00:00:57.000
I wrote it this way because
we're just going to be dealing

00:00:57.000 --> 00:00:59.360
with rows in this video and it
makes our notation a little

00:00:59.360 --> 00:01:01.600
bit easier.

00:01:01.600 --> 00:01:05.890
Let me focus on these
two rows right here.

00:01:05.890 --> 00:01:11.630
And let me define another
matrix B that is

00:01:11.630 --> 00:01:13.920
also an n by n matrix.

00:01:13.920 --> 00:01:19.510
And it's identical to matrix
A except for one row.

00:01:19.510 --> 00:01:21.720
So it's identical to matrix
A except for one row.

00:01:21.720 --> 00:01:23.430
You have r1 just like
that; it's the

00:01:23.430 --> 00:01:25.040
same as that one there.

00:01:25.040 --> 00:01:30.020
r2, keep going, go down to our
ri, even that one's identical.

00:01:30.020 --> 00:01:33.570
But rj I've now replaced.

00:01:33.570 --> 00:01:41.300
I'm replacing rj with rj minus
a scalar multiple of ri.

00:01:41.300 --> 00:01:45.110
Minus c times ri.

00:01:45.110 --> 00:01:46.970
So minus a scalar multiple
of that.

00:01:46.970 --> 00:01:48.730
I've replaced rj with that.

00:01:48.730 --> 00:01:52.340
So this is equivalent to the
row operations we do we did

00:01:52.340 --> 00:01:55.470
our Gaussian Elimination, or
when we put things in reduced

00:01:55.470 --> 00:01:56.690
row echelon form.

00:01:56.690 --> 00:02:00.930
And everything else in this
matrix is the same as A.

00:02:00.930 --> 00:02:02.990
It's all the way down to rn.

00:02:07.160 --> 00:02:09.610
This is our matrix B.

00:02:09.610 --> 00:02:12.900
So let's think about what the
determinant of B is going to

00:02:12.900 --> 00:02:14.330
be equal to.

00:02:14.330 --> 00:02:15.580
I'll do it in blue.

00:02:19.780 --> 00:02:24.330
Well, you could immediately say
that B is equivalent to--

00:02:24.330 --> 00:02:26.216
Well, you can imagine
two vectors.

00:02:29.060 --> 00:02:31.150
You can imagine two matrices.

00:02:31.150 --> 00:02:33.790
One matrix that look
like this.

00:02:33.790 --> 00:02:41.800
One matrix that look like r1,
r2, all the way down ri, all

00:02:41.800 --> 00:02:44.110
the way down to rj.

00:02:44.110 --> 00:02:46.140
And then you keep going
down to rn.

00:02:46.140 --> 00:02:48.720
That's one matrix, which you
may have already noticed is

00:02:48.720 --> 00:02:49.920
identical to A.

00:02:49.920 --> 00:02:50.820
That's one matrix.

00:02:50.820 --> 00:02:55.320
Then you could have another
matrix here

00:02:55.320 --> 00:02:58.050
that looks like this.

00:02:58.050 --> 00:02:59.700
It's identical everywhere.

00:02:59.700 --> 00:03:04.540
r1, r2, ri.

00:03:04.540 --> 00:03:07.350
Some dots there to show you I
might have skipped some rows.

00:03:07.350 --> 00:03:09.150
Skip some more rows.

00:03:09.150 --> 00:03:17.350
And then you have c
times times ri.

00:03:17.350 --> 00:03:18.710
c times ri.

00:03:18.710 --> 00:03:19.760
Let me do that in a
different color.

00:03:19.760 --> 00:03:22.130
This is ri right here.

00:03:24.800 --> 00:03:29.550
And then you just keep
going down to rn.

00:03:29.550 --> 00:03:34.490
Now, the determinant of B, you
could view as the determinant

00:03:34.490 --> 00:03:35.120
of this guy.

00:03:35.120 --> 00:03:36.120
Let me write this here.

00:03:36.120 --> 00:03:42.360
The determinant of B is equal to
the determinant of this guy

00:03:42.360 --> 00:03:45.200
plus the determinant
of this guy.

00:03:45.200 --> 00:03:48.150
Hopefully, you remember a couple
of videos ago, that if

00:03:48.150 --> 00:03:51.400
one matrix-- Let's have two
matrices that are identical in

00:03:51.400 --> 00:03:54.880
every way except for one row.

00:03:54.880 --> 00:03:57.850
So these two matrices are
completely identical except

00:03:57.850 --> 00:04:00.380
for what's going on
on the jth row.

00:04:00.380 --> 00:04:02.210
Here you have a r-sub-j.

00:04:02.210 --> 00:04:04.790
Here you have a c
times r-sub-i.

00:04:04.790 --> 00:04:07.990
So it's a scalar multiple
of a row that you had

00:04:07.990 --> 00:04:09.460
up here, this guy.

00:04:09.460 --> 00:04:12.610
So this is ri, this
is the ith row.

00:04:12.610 --> 00:04:14.350
Here you have an ri, here
you have an ri.

00:04:14.350 --> 00:04:16.850
But here you have another
version of r row, scalar

00:04:16.850 --> 00:04:19.610
multiple of ri, while
here you have an rj.

00:04:19.610 --> 00:04:22.610
Now, if you have another matrix
that is essentially

00:04:22.610 --> 00:04:26.300
identical to these two
matrices, except

00:04:26.300 --> 00:04:27.710
for this one row.

00:04:27.710 --> 00:04:31.130
And in that one row, it looks
like the addition of these two

00:04:31.130 --> 00:04:35.210
matrices-- and let me
put a negative here.

00:04:35.210 --> 00:04:38.320
So if you kept this matrix
completely identical, but if

00:04:38.320 --> 00:04:41.690
you were to replace it with
the sum of these two rows.

00:04:41.690 --> 00:04:46.750
So rj minus c times ri, you'll
get this matrix right here.

00:04:46.750 --> 00:04:47.910
You'll get matrix B.

00:04:47.910 --> 00:04:50.360
And we learned that the
determinant of B is equal to

00:04:50.360 --> 00:04:52.620
the determinant of this
guy and that guy.

00:04:52.620 --> 00:04:54.850
Remember, B is not the sum
of these two matrices.

00:04:54.850 --> 00:04:58.810
B is identical to these two
matrices, except for that one

00:04:58.810 --> 00:05:03.190
row where B's jth row is
equivalent to the jth row of

00:05:03.190 --> 00:05:06.870
this guy, plus the jth
row of that guy.

00:05:06.870 --> 00:05:09.220
And when I talk about adding
rows, you're just adding their

00:05:09.220 --> 00:05:10.840
corresponding elements.

00:05:10.840 --> 00:05:20.430
So I could rewrite this so this
row would look like-- The

00:05:20.430 --> 00:05:28.650
first term would be aj1
minus c times ai1.

00:05:28.650 --> 00:05:31.250
That would be the first
term in that row.

00:05:31.250 --> 00:05:34.010
The second term of that
row would be aj2

00:05:34.010 --> 00:05:36.650
minus c times ai2.

00:05:36.650 --> 00:05:40.570
And it would go all the
way to ajn minus

00:05:40.570 --> 00:05:45.820
ca-sub-in, the nth column.

00:05:45.820 --> 00:05:47.480
So that's all it
means by that.

00:05:47.480 --> 00:05:49.430
So the determinant of B is equal
to the determinant of

00:05:49.430 --> 00:05:52.260
this plus the determinant
of this.

00:05:52.260 --> 00:05:54.420
The determinant of this,
this thing right here

00:05:54.420 --> 00:05:55.470
is our matrix A.

00:05:55.470 --> 00:05:57.220
This is going to be the
determinant of A.

00:06:01.320 --> 00:06:03.540
And what's the determinant
of this?

00:06:03.540 --> 00:06:05.950
Well, let's break this down
a little bit more.

00:06:05.950 --> 00:06:10.350
The determine of this
is equal to what?

00:06:10.350 --> 00:06:14.025
This is completely equivalent to
A, except one of its rows--

00:06:14.025 --> 00:06:18.570
Sorry, this is completely
equivalent to this matrix.

00:06:18.570 --> 00:06:19.670
Not equivalent to A.

00:06:19.670 --> 00:06:20.620
Be very careful.

00:06:20.620 --> 00:06:22.810
Don't listen to everything
I say.

00:06:22.810 --> 00:06:25.070
It's not equivalent to A.

00:06:25.070 --> 00:06:27.340
The difference is that
A has an rj here.

00:06:27.340 --> 00:06:30.740
This guy has a minus
c times ri.

00:06:30.740 --> 00:06:35.860
So this is equivalent
to this matrix.

00:06:35.860 --> 00:06:39.560
It's completely equivalent to
this matrix right here.

00:06:39.560 --> 00:06:41.080
Let me do it like this.

00:06:41.080 --> 00:06:48.510
So you have an r1, r2, keep
going, and you have an ri,

00:06:48.510 --> 00:06:50.050
then you have another ri.

00:06:50.050 --> 00:06:51.630
Let me clean this
up a little bit.

00:06:51.630 --> 00:06:55.030
Let me clear this out
just so I have some

00:06:55.030 --> 00:06:56.280
space to work with.

00:06:59.230 --> 00:07:00.330
You have an ri.

00:07:00.330 --> 00:07:01.430
You have that ri there.

00:07:01.430 --> 00:07:03.740
Then you have another ri.

00:07:03.740 --> 00:07:06.200
You have another
ri right there.

00:07:06.200 --> 00:07:08.290
You have another ri.

00:07:08.290 --> 00:07:11.760
So the jth row has
an ri there.

00:07:11.760 --> 00:07:14.900
Then you keep going and
you have an r-sub-n.

00:07:14.900 --> 00:07:19.320
These two guys are completely
equivalent except for this guy

00:07:19.320 --> 00:07:23.840
has a minus c times
the jth row.

00:07:23.840 --> 00:07:24.490
Right?

00:07:24.490 --> 00:07:25.850
That's what this was,
right here.

00:07:25.850 --> 00:07:26.820
This is the jth row.

00:07:26.820 --> 00:07:28.420
Everything we're doing
is in the jth row.

00:07:28.420 --> 00:07:30.520
This has a minus c times
the jth row.

00:07:33.740 --> 00:07:37.005
So the determinant of this guy
right here-- Let me just be

00:07:37.005 --> 00:07:38.520
clear that I'm only taking
the determinant of

00:07:38.520 --> 00:07:39.970
this guy right here.

00:07:39.970 --> 00:07:46.450
It's going to be equal to minus
c times the determinant

00:07:46.450 --> 00:07:49.640
of-- let me write it this
way --minus c times the

00:07:49.640 --> 00:07:55.600
determinant of r1, r2.

00:07:55.600 --> 00:07:59.170
You have your first ri.

00:07:59.170 --> 00:08:00.640
And then in the jth row
you have another

00:08:00.640 --> 00:08:02.190
version of the ri.

00:08:02.190 --> 00:08:04.480
And then you go down
to r-sub-n.

00:08:04.480 --> 00:08:06.410
So times that determinant.

00:08:06.410 --> 00:08:07.910
This is just the determinant
of this.

00:08:07.910 --> 00:08:10.920
I've added brackets and
straight lines.

00:08:10.920 --> 00:08:12.890
And we saw this a couple
of videos ago.

00:08:12.890 --> 00:08:17.255
If you have a matrix, you just
multiply one of its rows by a

00:08:17.255 --> 00:08:18.740
scalar, in this case minus c.

00:08:18.740 --> 00:08:20.510
It's equivalent to minus c.

00:08:20.510 --> 00:08:23.280
The determinant of the new
matrix is equal to minus c

00:08:23.280 --> 00:08:25.160
times the determinant
of your matrix.

00:08:25.160 --> 00:08:26.810
That's all I'm saying
right here.

00:08:26.810 --> 00:08:29.760
But what is the determinant
of this matrix?

00:08:29.760 --> 00:08:32.799
You might have already noticed
that it has duplicate rows.

00:08:32.799 --> 00:08:38.440
It has an ri, and then in the
ith row, then it has another

00:08:38.440 --> 00:08:42.090
ri in the jth row.

00:08:42.090 --> 00:08:47.010
Remember, we kind of decomposed
this B matrix right

00:08:47.010 --> 00:08:49.630
here as the sum of-- Or its
determinant can be described

00:08:49.630 --> 00:08:51.930
as the determinant of the
sum of these two things.

00:08:51.930 --> 00:08:54.400
B isn't the sum of
these two things.

00:08:54.400 --> 00:08:56.500
Every other element is identical
to every other

00:08:56.500 --> 00:08:58.410
element in each of these guys.

00:08:58.410 --> 00:09:02.100
But this guy right here,
he has duplicate ri's.

00:09:02.100 --> 00:09:04.580
And what do we know about the
determinant of a matrix with

00:09:04.580 --> 00:09:06.110
duplicate entries?

00:09:06.110 --> 00:09:08.260
The determinant is zero.

00:09:08.260 --> 00:09:10.420
So this entry right
here is zero.

00:09:10.420 --> 00:09:12.620
Minus c times 0, 0.

00:09:12.620 --> 00:09:15.570
So the determinant of this
whole thing is 0.

00:09:15.570 --> 00:09:18.430
So the big take-away right here
is that the determinant

00:09:18.430 --> 00:09:20.930
of B is equal to just the
determinant of this thing,

00:09:20.930 --> 00:09:23.090
which was the determinant
of A.

00:09:23.090 --> 00:09:24.590
This is a very big take-away.

00:09:24.590 --> 00:09:25.890
It's going to make our
life very easy.

00:09:25.890 --> 00:09:31.300
The determinant of B is equal
to the determinant of A.

00:09:31.300 --> 00:09:34.780
So if you start with some
matrix, and you replace the

00:09:34.780 --> 00:09:36.700
jth row in this example,
but any row.

00:09:36.700 --> 00:09:42.060
If you replace any row with
that row minus some scalar

00:09:42.060 --> 00:09:45.020
multiple of another row-- we
picked ri in this case, that

00:09:45.020 --> 00:09:50.560
would be ri --the determinant
will not be changed.

00:09:50.560 --> 00:09:52.500
You have to be very particular
about how you say it because,

00:09:52.500 --> 00:09:54.500
obviously, if you just
multiplied something by a

00:09:54.500 --> 00:09:56.690
scalar-- if you were to change
its determinant, or if you do

00:09:56.690 --> 00:09:57.700
other things.

00:09:57.700 --> 00:10:01.390
If you just take a row, if you
take the jth row, and you

00:10:01.390 --> 00:10:06.270
replace it with the jth row
minus c times the ith row

00:10:06.270 --> 00:10:10.140
times some other row, which is
equivalent to just a row

00:10:10.140 --> 00:10:14.010
operation that we have been
doing, then it will not change

00:10:14.010 --> 00:10:15.060
your determinant.

00:10:15.060 --> 00:10:17.670
Which is a very big take-away
because now we can carefully

00:10:17.670 --> 00:10:22.210
do some row operations and know
that the determinant will

00:10:22.210 --> 00:10:23.860
not change.