n x n determinant

WEBVTT
Kind: captions
Language: en

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So far we've been able to define
the determinant for a

00:00:03.380 --> 00:00:04.600
2-by-2 matrix.

00:00:04.600 --> 00:00:08.700
This was our definition right
here: ad minus bc.

00:00:08.700 --> 00:00:10.960
And then we were able to broaden
that a bit by creating

00:00:10.960 --> 00:00:14.460
a definition for the determinant
of a 3-by-3

00:00:14.460 --> 00:00:19.330
matrix, and we did that right
here, where we essentially

00:00:19.330 --> 00:00:22.190
said the determinant is equal
to each of these terms-- you

00:00:22.190 --> 00:00:25.490
could call these maybe the
coefficient terms-- times the

00:00:25.490 --> 00:00:28.682
determinant of the matrix-- you
can kind of view it as the

00:00:28.682 --> 00:00:31.450
submatrix produced-- when you
get rid of each of these guys'

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column and row.

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So when you got rid of this
guy's column and row, you're

00:00:35.340 --> 00:00:36.480
left with this matrix.

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So we said this guy times
the determinant of this.

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And we kept switching signs,
minus this guy times the

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determinant, if you move
his column and his row.

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So it was left with these terms
right there to get that

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determinant.

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Then finally, you switched
signs again.

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So plus this guy times the
determinant of the 2-by-2

00:00:54.620 --> 00:00:56.860
matrix if you get rid of this
row and this column.

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So this thing right here,
which was this matrix.

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Now let's see if we can
extend this to a

00:01:01.850 --> 00:01:03.980
general n-by-n matrix.

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So let's write out our n-by-n
matrix right over here.

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I'll do it in blue.

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So let's say I have some matrix
A that is an n-by-n

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matrix, so it's going
to look like this.

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This would be a11, that would
be a12, and we would go all

00:01:23.870 --> 00:01:28.270
the way to-- you're going
to have n columns, a1n.

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And when you go down, this is
going to be your second row:

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a21, and it's going to go all
the way down to an1, because

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you have n rows as well.

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And then if you go down the
diagonal all the way, this

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right here would be ann.

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So there is my n-by-n matrix.

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Now, before I define how to find
the determinant of this,

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let me make another
definition.

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Let me define-- so this
is my matrix A.

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Let me define a submatrix Aij to
be equal to-- see this is n

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by n, right?

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So this is going to be an n
minus 1 by n minus 1 matrix.

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So if this is 7 by 7, the
submatrix is going to be 6 by

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6, one less in each direction.

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So this is going to be the n
minus 1 by n minus 1 matrix

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you get if you essentially
ignore or if you take away--

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maybe I should say take away.

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Let's say ignore, like
the word ignore.

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If you ignore the i-th row, this
right here is the row,

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the i-th row and the
j-th column of a.

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So, for example, let's go back
to our 3 by 3 right here.

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This thing could be denoted
based on that definition I--

00:03:03.370 --> 00:03:06.670
we could have called
this, this was a11,

00:03:06.670 --> 00:03:08.000
this term right here.

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We could denote the matrix when
you get rid of the first

00:03:11.620 --> 00:03:13.680
column and the first row or the
first row and the first

00:03:13.680 --> 00:03:16.750
column, we could call this thing
right here, we could

00:03:16.750 --> 00:03:19.550
call that big matrix A11.

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So this was big matrix A11.

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This is big matrix A21, or
actually, this matrix was

00:03:28.780 --> 00:03:32.980
called C, so this would
be C11 right there.

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We could call this one, this
would be matrix C12.

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Why is that?

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Because if you get rid of the
first row, let me get rid of

00:03:45.520 --> 00:03:47.610
the first row, right?

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The first term is your row.

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If you get rid of the first
row and the second column,

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this is the matrix that's
left over: 2, 3, 4, 1.

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So this is this guy
and this guy.

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2, 3, 4, 1.

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So this is the submatrix c
because this is the big matrix

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C, But this one is C12.

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I know it's a little
bit messy there.

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So that's all we mean
by the submatrix.

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Very similar to what we did
in the 3 by 3 case.

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You essentially get rid of--
so if you want to find out

00:04:21.000 --> 00:04:23.970
this guy's submatrix, you would
call that a11, and you

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would literally cross out the
first row and the first

00:04:26.630 --> 00:04:29.120
column, and everything
left over here

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would be that submatrix.

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Now, with that out of the way,
we can create a definition,

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and it might seem a little
circular to you at first, and

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on some level it is.

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We're going to define the
determinant of A to be equal

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to-- this is interesting.

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It's actually a recursive
definition.

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I'll talk about that
in a second.

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It's equal to-- we start
with a plus.

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It's equal to a11 times the
submatrix if you remove this

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guy's row and column.

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So that, by definition, is
just A, big capital A11's

00:05:05.940 --> 00:05:07.460
determinant.

00:05:07.460 --> 00:05:08.710
So that's exactly what we did.

00:05:08.710 --> 00:05:10.540
Let me write that a
little bit neater.

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So times the determinant of
its submatrix, so the

00:05:14.670 --> 00:05:17.550
determinant of A11.

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So you take A11, you get rid of
its column and its row or

00:05:20.416 --> 00:05:22.740
its row and its column, and
everything else, you find the

00:05:22.740 --> 00:05:24.120
determinant of that.

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Actually, let me write it
in terms of-- let me

00:05:25.854 --> 00:05:27.600
write it this way.

00:05:27.600 --> 00:05:33.420
a11 times the determinant
of the submatrix A11.

00:05:33.420 --> 00:05:35.130
And then we switch sides.

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We're just going to go along
this row, and then you do

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minus a12 times the determinant
of its submatrix,

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which we'll just call A12.

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We would get rid of this row
and this column, and

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everything left would
be this matrix A12.

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We want to find its
determinant.

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And then we'll take the next
guy over here would be a13.

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So we switch signs with minus.

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Now, you go plus,
so a13 times the

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determinant of its submatrix.

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So if this is n by n, these each
are going to be n minus 1

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by n minus 1.

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So the determinant of A13.

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And you're just going to keep
doing that, keep switching

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signs, so it's going to be a
minus and then a plus and you

00:06:19.020 --> 00:06:21.550
keep going all the way--
and then I don't know.

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It depends on whether an,
whether we're dealing with an

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odd number or an even number.

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If we're dealing with an even
number, this is going to be a

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minus sign.

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If it's an odd number, it's
going to be a plus sign, but

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you get the idea.

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It's either going to be a plus
or a minus, not just-- if it's

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odd, this is going
to be a plus.

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If it's an even n, it's going to
be a minus, All the way to

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a1n, the n-th column times
its submatrix, A1n.

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With that submatrix, you get rid
of the first row and the

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n-th column, and it's going
to be everything

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that's left in between.

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And you immediately might
say, Sal, what kind of a

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definition is this?

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You defined a determinant for an
arbitrary n-by-n matrix in

00:07:01.130 --> 00:07:03.680
terms of another definition
of a determinant.

00:07:03.680 --> 00:07:04.880
How does this work?

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And the reason why this works
is because the determinant

00:07:09.330 --> 00:07:11.630
that you use in the definition
are determinants

00:07:11.630 --> 00:07:13.090
of a smaller matrix.

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So this is a determinant of an n
minus 1 by n minus 1 matrix.

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And you're saying hey, Sal, that
still doesn't make any

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sense because we don't know how
to find the determinant of

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an n minus 1 by n
minus 1 matrix.

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Well, you apply this definition
again, and then

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it's going to be in terms of n
minus 2 times n-- or n minus 2

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by n minus 2 matrices.

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And you're like how
do you do that?

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Well, you keep doing it, and
you're going to get all the

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way down to a 2-by-2 matrix.

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And that one we defined well.

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We defined the determinant of
a 2-by-2 matrix not in terms

00:07:42.640 --> 00:07:43.240
of a determinant.

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We just defined it in terms of
a times-- we defined it as--

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let me write it up here.

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It was a times d minus
b times c.

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And you can see.

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I mean, we could just go down to
the 3 by 3, but the 2 by 2

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is really the most fundamental
definition.

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And you could see that the
definition of a 3-by-3

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determinant is a special
case of the general

00:08:02.720 --> 00:08:04.110
case for an n by n.

00:08:04.110 --> 00:08:07.040
We take this guy and we
multiply him times the

00:08:07.040 --> 00:08:09.800
determinant of his submatrix
right there.

00:08:09.800 --> 00:08:11.810
Then we take this guy where
we switch signs.

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We have a minus.

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And we multiply him times the
determinant of his submatrix,

00:08:15.700 --> 00:08:17.520
which is that right there.

00:08:17.520 --> 00:08:18.950
Then you do a plus.

00:08:18.950 --> 00:08:21.455
You switch signs and then you
multiply this guy times the

00:08:21.455 --> 00:08:24.030
determinant of his submatrix,
which is that right there.

00:08:24.030 --> 00:08:26.870
So this is a general case
of what I just defined.

00:08:26.870 --> 00:08:30.040
But we know it's never that
satisfying to deal in the

00:08:30.040 --> 00:08:32.520
abstract or the generalities.

00:08:32.520 --> 00:08:33.900
We want to do a specific case.

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And actually, before I do that,
let me just introduce a

00:08:36.190 --> 00:08:36.610
term to you.

00:08:36.610 --> 00:08:40.120
This is called a recursive
formula.

00:08:40.120 --> 00:08:42.090
And if you become a computer
science major,

00:08:42.090 --> 00:08:43.120
you'll see this a lot.

00:08:43.120 --> 00:08:47.150
But a recursive function or a
recursive formula is defined

00:08:47.150 --> 00:08:48.630
in terms of itself.

00:08:48.630 --> 00:08:51.110
But the things that you use in
the definition use a slightly

00:08:51.110 --> 00:08:54.480
simpler version of it, and as
you keep going through, or you

00:08:54.480 --> 00:08:56.730
keep recursing through it, you
get simpler and simpler

00:08:56.730 --> 00:08:59.480
versions of it until you get
to some type of base case.

00:08:59.480 --> 00:09:03.360
In this case, our base case is
the case of a 2-by-2 matrix.

00:09:03.360 --> 00:09:05.030
You keep doing this, and
eventually you'll get to a

00:09:05.030 --> 00:09:07.220
determinant of a 2-by-2 matrix,
and we know how to

00:09:07.220 --> 00:09:08.260
find those.

00:09:08.260 --> 00:09:09.735
So this is a recursive
definition.

00:09:13.540 --> 00:09:15.930
But let's actually apply it
because I think that's what

00:09:15.930 --> 00:09:18.960
actually makes things
concrete.

00:09:18.960 --> 00:09:23.140
So let's take-- this is going
to be computationally

00:09:23.140 --> 00:09:25.900
intensive, but I think if we
focus, we can get there.

00:09:25.900 --> 00:09:32.070
So I'm going to have a 4-by-4
matrix: 1, 2, 3, 4.

00:09:32.070 --> 00:09:35.080
1-- throw some zeroes in there
to make the computation a

00:09:35.080 --> 00:09:43.400
little bit simpler, 0, 1, 2,
3, and then 2, 3, 0, 0.

00:09:43.400 --> 00:09:48.640
So let's figure out this
determinant right there.

00:09:48.640 --> 00:09:50.040
This is the determinant
of the matrix.

00:09:50.040 --> 00:09:51.260
If I put some brackets
there that would

00:09:51.260 --> 00:09:52.100
have been the matrix.

00:09:52.100 --> 00:09:55.000
But let's find the determinant
of this matrix.

00:09:55.000 --> 00:09:58.590
So this is going to be equal
to-- by our definition, it's

00:09:58.590 --> 00:10:03.000
going to be equal to 1 times the
determinant of this matrix

00:10:03.000 --> 00:10:04.900
right here if you get rid of
this row and this column.

00:10:04.900 --> 00:10:10.220
So it's going to be 1 times the
determinant of 0, 2, 0; 1,

00:10:10.220 --> 00:10:13.320
2, 3; 3, 0, 0.

00:10:13.320 --> 00:10:17.600
That's just this guy right here,
this matrix right there.

00:10:17.600 --> 00:10:20.660
Then I'm going to have a 2, but
I'm going to switch signs.

00:10:20.660 --> 00:10:24.580
So it's minus 2 times the
determinant if I get rid of

00:10:24.580 --> 00:10:27.880
that row and this column,
so it's 1, 2, 0.

00:10:27.880 --> 00:10:30.130
I'm ignoring the zero because
it's in the same column as the

00:10:30.130 --> 00:10:40.525
2: 1, 2, 0; 0, 2, 3,
and then 2, 0, 0.

00:10:43.330 --> 00:10:44.540
And then I switch signs again.

00:10:44.540 --> 00:10:47.240
It was a minus, so now
I go back to plus.

00:10:47.240 --> 00:10:50.720
So I do that guy, so
plus 3 times the

00:10:50.720 --> 00:10:52.040
determinant of his submatrix.

00:10:52.040 --> 00:10:53.330
Get rid of that row and get
rid of that columm,

00:10:53.330 --> 00:10:54.766
I get a 1, 0, 0.

00:10:57.600 --> 00:11:00.010
I get a 0, 1, 3.

00:11:05.100 --> 00:11:06.700
I skip this column every time.

00:11:06.700 --> 00:11:15.010
Then I get a 2, 3, 0,
just like that.

00:11:15.010 --> 00:11:16.210
We're almost done.

00:11:16.210 --> 00:11:18.330
One more in this column.

00:11:18.330 --> 00:11:21.290
Let me switch to
another color.

00:11:21.290 --> 00:11:23.160
I haven't used the
blue in this yet.

00:11:23.160 --> 00:11:25.110
So then I'm going
to do a minus 4.

00:11:25.110 --> 00:11:29.870
Remember, it's plus, minus,
plus, minus 4 times the

00:11:29.870 --> 00:11:31.130
determinant of its submatrix.

00:11:31.130 --> 00:11:33.480
That's going to be
that right there.

00:11:33.480 --> 00:11:42.330
So it's 1, 0, 2; 0, 1, 2;
2, 3, 0, just like that.

00:11:42.330 --> 00:11:44.040
And now we're down to
the 3-by-3 case.

00:11:44.040 --> 00:11:46.390
We could use the definition of
the 3 by 3, but we could just

00:11:46.390 --> 00:11:47.980
keep applying this recursive
definition.

00:11:51.310 --> 00:11:56.140
So this is going to be equal
to-- let me write it here.

00:11:56.140 --> 00:11:59.440
It's 1 times-- what's
this determinant?

00:11:59.440 --> 00:12:03.640
This determinant's going to be
0 times the determinant of

00:12:03.640 --> 00:12:05.950
that submatrix, 2, 3, 0, 0.

00:12:08.900 --> 00:12:11.150
That was this one right here.

00:12:11.150 --> 00:12:15.110
And then we have minus 2, minus
this 2-- remember, we

00:12:15.110 --> 00:12:18.000
switched signs-- plus, minus,
plus, so minus 2 times its

00:12:18.000 --> 00:12:21.020
submatrix, so it's 1, 3, 3, 0.

00:12:26.340 --> 00:12:31.650
And then finally plus 0 times
its submatrix, which is this

00:12:31.650 --> 00:12:39.370
thing right here: 1, 2,
3, 0, just like that.

00:12:39.370 --> 00:12:41.770
And then we have this
next guy right here.

00:12:41.770 --> 00:12:45.060
As you can see, this can get a
little bit tedious, but we'll

00:12:45.060 --> 00:12:46.130
keep our spirits up.

00:12:46.130 --> 00:12:53.570
So minus 2 times 1 times its
submatrix, so that's this guy

00:12:53.570 --> 00:12:56.850
right here-- times the
determinant of its submatrix

00:12:56.850 --> 00:12:59.840
2, 3, 0, 0.

00:12:59.840 --> 00:13:04.290
Then minus 2 times-- get
rid of that row, that

00:13:04.290 --> 00:13:06.190
column-- 0, 3, 2, 0.

00:13:09.320 --> 00:13:13.620
And then plus 0 times
0, 2, 2, 0.

00:13:17.060 --> 00:13:19.730
That's this one right there.

00:13:19.730 --> 00:13:22.510
Halfway there, at
least for now.

00:13:22.510 --> 00:13:28.270
And then we get this next one,
so we have a plus 3.

00:13:28.270 --> 00:13:29.770
Bring out our parentheses.

00:13:29.770 --> 00:13:32.570
And then we're going to have 1
times its sub-- I guess call

00:13:32.570 --> 00:13:33.820
it sub-determinant.

00:13:33.820 --> 00:13:39.580
So 1 times the determinant
1, 3, 3, 0, right?

00:13:39.580 --> 00:13:41.710
You get rid of this guy's row
and column, you get this guy

00:13:41.710 --> 00:13:42.870
right there.

00:13:42.870 --> 00:13:48.620
And then minus 0-- get rid
of this row and column--

00:13:48.620 --> 00:13:50.246
times 0, 3, 2, 0.

00:13:53.300 --> 00:13:59.880
Then you have plus 0 times its
sub-determinant 0, 1, 2, 3.

00:14:02.740 --> 00:14:04.230
Three-fourths of
the way there.

00:14:04.230 --> 00:14:05.180
One last term.

00:14:05.180 --> 00:14:07.790
Let's hope we haven't made
any careless mistakes.

00:14:07.790 --> 00:14:15.470
Minus 4 times 1 times 1,
2, 3, 0 right there.

00:14:18.810 --> 00:14:23.415
Minus 0 times-- get rid of those
two guys-- 0, 2, 2, 0.

00:14:26.560 --> 00:14:30.870
And then plus 2 times
0, 1, 2, 3, right?

00:14:30.870 --> 00:14:34.130
Plus 2-- get rid of these
guys-- 0, 1, 2, 3.

00:14:37.660 --> 00:14:41.350
Now, we've defined or we've
calculated or we've defined

00:14:41.350 --> 00:14:44.060
our determinant of this matrix
in terms of just a bunch of

00:14:44.060 --> 00:14:45.230
2-by-2 matrices.

00:14:45.230 --> 00:14:48.170
So hopefully, you saw in this
example that the recursion

00:14:48.170 --> 00:14:48.680
worked out.

00:14:48.680 --> 00:14:51.410
So let's actually find what
this number is equal to.

00:14:51.410 --> 00:14:53.570
A determinant is always just
going to be a number.

00:14:53.570 --> 00:14:56.090
So let me get a nice
vibrant color.

00:14:56.090 --> 00:14:57.610
This is 0 times--
I don't care.

00:14:57.610 --> 00:14:59.940
0 times anything's
going to be 0.

00:14:59.940 --> 00:15:02.220
0 times anything is
going to be 0.

00:15:02.220 --> 00:15:04.550
0 times anything's
going to be 0.

00:15:04.550 --> 00:15:06.460
0 times anything's
going to be 0.

00:15:06.460 --> 00:15:07.470
Just simplifying it.

00:15:07.470 --> 00:15:10.420
These guys are 0 because
it's 0 times that.

00:15:10.420 --> 00:15:12.910
0 times this is going
to be equal to 0.

00:15:12.910 --> 00:15:14.160
So what are we left with?

00:15:17.100 --> 00:15:22.090
This is going to be equal to 1
times-- this is all we have

00:15:22.090 --> 00:15:25.250
left here is a minus 2 times--
and what is this determinant?

00:15:25.250 --> 00:15:28.450
It's 1 times 0, which is 0.

00:15:28.450 --> 00:15:30.220
It's 0-- let me write this.

00:15:30.220 --> 00:15:34.870
This is going to be 1 times 0
is 0, minus 3 times 3 is 0

00:15:34.870 --> 00:15:37.980
minus 9, so minus 9.

00:15:37.980 --> 00:15:39.940
This right here is
just minus 9.

00:15:39.940 --> 00:15:42.270
So minus 2 times minus 9.

00:15:42.270 --> 00:15:44.840
That's our first thing, I'll
simplify it in a second.

00:15:44.840 --> 00:15:47.030
Now let's do this
term right here.

00:15:47.030 --> 00:15:51.060
So it's minus 2 times-- now
what's this determinant?

00:15:51.060 --> 00:15:54.280
2 times 0 minus 0 times 3.

00:15:54.280 --> 00:15:55.030
That's 0 minus 0.

00:15:55.030 --> 00:15:57.040
So this is 0.

00:15:57.040 --> 00:15:59.990
That guy became 0, so we
can ignore that term.

00:15:59.990 --> 00:16:03.050
This term right here is
0 times 0, which is 0,

00:16:03.050 --> 00:16:04.370
minus 2 times 3.

00:16:04.370 --> 00:16:06.210
So it's minus 6.

00:16:06.210 --> 00:16:10.780
So it's minus 2 times-- so this
is a minus 6 right here.

00:16:10.780 --> 00:16:15.150
You have a minus 2 times a minus
6, so that's a plus 12.

00:16:15.150 --> 00:16:16.770
So I'll just write
a plus 12 here.

00:16:16.770 --> 00:16:19.910
This minus 2 is that minus
2 right there.

00:16:19.910 --> 00:16:23.370
And then we have a plus 3.

00:16:23.370 --> 00:16:27.020
And then this first term is 1
times 0, which is 0, minus--

00:16:27.020 --> 00:16:29.880
let me make the parentheses
here-- 1 times 0, which is 0,

00:16:29.880 --> 00:16:33.395
minus 3 times 3, which
is minus 9 times 1.

00:16:33.395 --> 00:16:34.840
So it's minus 9.

00:16:34.840 --> 00:16:37.120
Everything else was a 0.

00:16:37.120 --> 00:16:38.780
We're in the home stretch.

00:16:38.780 --> 00:16:41.750
We have a minus 4.

00:16:41.750 --> 00:16:47.670
Let's see, this is 1 times 0,
which is 0, minus 3 times 2,

00:16:47.670 --> 00:16:49.100
so minus 6.

00:16:49.100 --> 00:16:51.660
So this is minus 6 right here.

00:16:51.660 --> 00:16:54.950
Minus 6, this is 0, and then we
have this guy right here.

00:16:54.950 --> 00:16:59.290
So we have 0 times 3, which
is 0, minus 2 times 1.

00:16:59.290 --> 00:17:03.350
So that's minus 2, and then
you have a minus 2 times a

00:17:03.350 --> 00:17:06.440
plus 2 is minus 4.

00:17:06.440 --> 00:17:08.250
So now we just have
to make sure we do

00:17:08.250 --> 00:17:10.900
our arithmetic properly.

00:17:10.900 --> 00:17:15.290
This is 1 times plus 18,
so this is 18, right?

00:17:15.290 --> 00:17:17.230
Minus 2 times minus 9.

00:17:17.230 --> 00:17:21.819
This right here is minus 24.

00:17:21.819 --> 00:17:27.010
This right here is minus 27.

00:17:27.010 --> 00:17:30.270
And then this right here, let's
see, this is minus 10

00:17:30.270 --> 00:17:31.030
right here.

00:17:31.030 --> 00:17:32.650
That is minus 10.

00:17:32.650 --> 00:17:39.860
Minus 4 times minus
10 is plus 40.

00:17:39.860 --> 00:17:42.950
Let's see if we can simplify
this a little bit.

00:17:42.950 --> 00:17:45.450
If we simplify this a little
bit-- I don't want to make a

00:17:45.450 --> 00:17:47.540
careless mistake right
at the end.

00:17:47.540 --> 00:17:54.690
So 18 minus 24, 24 minus 18 is
6, so this is going to be

00:17:54.690 --> 00:17:57.530
equal to minus 6, right?

00:17:57.530 --> 00:18:00.090
18 minus 24 is minus 6.

00:18:00.090 --> 00:18:03.270
And then-- let me do it in
green-- now what's the

00:18:03.270 --> 00:18:03.740
difference?

00:18:03.740 --> 00:18:10.470
If we have minus 27 plus
40, that's 13, right?

00:18:10.470 --> 00:18:12.510
It's positive 13.

00:18:12.510 --> 00:18:18.310
So minus 6 plus positive
13 is equal to 7.

00:18:18.310 --> 00:18:19.460
And so we are done!

00:18:19.460 --> 00:18:21.700
After all of that computation,
hopefully we haven't made a

00:18:21.700 --> 00:18:22.970
careless mistake.

00:18:22.970 --> 00:18:25.030
The determinant of this
character right

00:18:25.030 --> 00:18:27.550
here is equal to 7.

00:18:27.550 --> 00:18:30.670
The determinant is equal to 7.

00:18:30.670 --> 00:18:33.310
And so the one useful takeaway,
we know that this is

00:18:33.310 --> 00:18:36.950
invertible because it has
a non-zero determinant.

00:18:36.950 --> 00:18:39.130
Hopefully, you found
that useful.


3 x 3 determinant

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:04.019
In the last video we defined the
notion of a determinant of

00:00:04.019 --> 00:00:05.160
a 2 by 2 matrix.

00:00:05.160 --> 00:00:09.490
So if I have some matrix-- let's
just call it B-- if my

00:00:09.490 --> 00:00:15.710
matrix B looks like this, if
its entries are a, b, c, d,

00:00:15.710 --> 00:00:20.490
we've defined to determinant
of B.

00:00:20.490 --> 00:00:23.910
Which could also be written as
B with these lines around it,

00:00:23.910 --> 00:00:27.840
which could also be written as
the entries of the matrix with

00:00:27.840 --> 00:00:30.690
those lines around
it, a, b, c, d.

00:00:30.690 --> 00:00:32.250
And I don't want you to
get these confused.

00:00:32.250 --> 00:00:34.420
This is the matrix when
you have the brackets.

00:00:34.420 --> 00:00:36.930
This is the determinant of the
matrix, when you just have

00:00:36.930 --> 00:00:38.360
these straight lines.

00:00:38.360 --> 00:00:44.760
And this, by definition, was
equal to ad minus bc.

00:00:44.760 --> 00:00:47.270
And you saw in the last video,
or maybe you saw in the last

00:00:47.270 --> 00:00:49.195
video, what the motivation
for this came from.

00:00:49.195 --> 00:00:53.010
When we figured out the inverse
of B, we determined

00:00:53.010 --> 00:01:02.500
that it was equal to 1 over ad
minus bc times another matrix,

00:01:02.500 --> 00:01:05.110
which was essentially these
two entry swaps, you

00:01:05.110 --> 00:01:06.340
got a d and an a.

00:01:06.340 --> 00:01:08.190
And then these two entries
made negative, so

00:01:08.190 --> 00:01:11.140
minus c and minus b.

00:01:11.140 --> 00:01:13.930
This was the inverse of b.

00:01:13.930 --> 00:01:16.560
And we said, well, when
is this defined?

00:01:16.560 --> 00:01:19.690
This is defined as long as this
character right here does

00:01:19.690 --> 00:01:20.850
not equal 0.

00:01:20.850 --> 00:01:23.150
So you said hey, this looks
pretty important.

00:01:23.150 --> 00:01:25.005
Let's call this thing right
there the determinant.

00:01:31.110 --> 00:01:40.000
And then we could say that B is
invertible, if and only if,

00:01:40.000 --> 00:01:46.010
the determinant of B
does not equal 0.

00:01:46.010 --> 00:01:50.230
Because if it equals 0, then
this formula for your inverse

00:01:50.230 --> 00:01:51.080
won't be well defined.

00:01:51.080 --> 00:01:53.970
And we just got this from our
technique of creating an

00:01:53.970 --> 00:01:55.870
augmented matrix whatnot.

00:01:55.870 --> 00:01:58.530
But the big take away is we
defined this notion of a

00:01:58.530 --> 00:02:01.120
determinant it for
a 2 by 2 matrix.

00:02:01.120 --> 00:02:04.120
Now the next question is, well
that's just a 2 by 2,

00:02:04.120 --> 00:02:06.950
everything we do in linear
algebra, we like to generalize

00:02:06.950 --> 00:02:09.930
it to higher numbers of
rows and columns.

00:02:09.930 --> 00:02:11.980
So the next step, at least--
let's just do baby steps--

00:02:11.980 --> 00:02:13.470
let's start with a 3 by 3.

00:02:13.470 --> 00:02:15.870
Let's define what its
determinant is.

00:02:15.870 --> 00:02:18.750
So let me construct a
3 by 3 matrix here.

00:02:18.750 --> 00:02:23.160
Let's say my matrix A is equal
to-- let me just write its

00:02:23.160 --> 00:02:28.100
entries-- first row, first
column, first row, second

00:02:28.100 --> 00:02:30.460
column, first row,
third column.

00:02:30.460 --> 00:02:34.920
Then you have a2
1, a2 2, a2 3.

00:02:34.920 --> 00:02:38.880
Then you have a3 1, third
row first column, a3

00:02:38.880 --> 00:02:41.570
2, and then a3 3.

00:02:41.570 --> 00:02:44.730
That is a 3 by 3 matrix.

00:02:44.730 --> 00:02:46.280
Three rows and three columns.

00:02:46.280 --> 00:02:48.660
This is 3 by 3.

00:02:48.660 --> 00:02:55.110
I am going to define the
determinant of A.

00:02:55.110 --> 00:02:56.630
So this is a definition.

00:02:56.630 --> 00:03:00.720
I'm going to define the
determinant of this 3 by 3

00:03:00.720 --> 00:03:04.010
matrix A as being equal to--
and this is a little bit

00:03:04.010 --> 00:03:06.110
convoluted, but you'll get the
hang of it eventually.

00:03:06.110 --> 00:03:07.890
In the next several videos we're
just going to do a ton

00:03:07.890 --> 00:03:08.350
of determinants.

00:03:08.350 --> 00:03:11.155
So it just becomes a bit of
second nature to you.

00:03:11.155 --> 00:03:13.770
It's a little computationally
intensive sometimes.

00:03:13.770 --> 00:03:16.600
But it equals this first row.

00:03:16.600 --> 00:03:22.080
It equals a1 1 times the
determinant of the matrix you

00:03:22.080 --> 00:03:26.020
get, if you get rid of this
guy's column and row.

00:03:26.020 --> 00:03:27.650
So if you get rid of this guy's
column and row, you're

00:03:27.650 --> 00:03:29.040
left with this matrix here.

00:03:29.040 --> 00:03:39.040
So times the determinant of the
matrix a2 2, a2 3, a3 2,

00:03:39.040 --> 00:03:41.140
and then a3 3.

00:03:41.140 --> 00:03:42.010
Just like that.

00:03:42.010 --> 00:03:45.490
So that's our first entry
and that's a plus this.

00:03:45.490 --> 00:03:47.680
And then I said it's a plus
this, because the next entry's

00:03:47.680 --> 00:03:49.330
going to be a minus.

00:03:49.330 --> 00:03:51.610
You have a minus this
guy right here.

00:03:51.610 --> 00:03:59.580
So then you're going to have
minus a1 2 times the matrix

00:03:59.580 --> 00:04:03.200
you get if you eliminate
his column and his row.

00:04:03.200 --> 00:04:06.210
So times, you're going to get
these entries right there.

00:04:06.210 --> 00:04:18.660
So a2 1, a2 3, a3 1, and
then you have a3 3.

00:04:18.660 --> 00:04:20.089
We're not quite done.

00:04:20.089 --> 00:04:21.839
You could probably guess with
the next one's going to be.

00:04:21.839 --> 00:04:25.550
Then you're going to have a
plus-- let me switch to a

00:04:25.550 --> 00:04:28.430
better color-- plus this guy.

00:04:28.430 --> 00:04:33.210
Plus a1 3 times the determinant
of its-- I guess

00:04:33.210 --> 00:04:34.850
you could call it--
its sub-matrix.

00:04:34.850 --> 00:04:35.930
We'll call it that for now.

00:04:35.930 --> 00:04:37.750
So this matrix right here.

00:04:37.750 --> 00:04:45.680
So a2 1, a2 2, a3 1, a3 2.

00:04:45.680 --> 00:04:47.130
This is our definition of the

00:04:47.130 --> 00:04:50.860
determinant of a 3 by 3 matrix.

00:04:50.860 --> 00:04:54.870
And the motivation is, because
when you take the determinant

00:04:54.870 --> 00:04:57.240
of a 3 by 3 it turns out-- I
haven't shown it to you yet--

00:04:57.240 --> 00:04:58.590
that the property is the same.

00:04:58.590 --> 00:05:01.410
That if the determinant of this
is 0, you will not be

00:05:01.410 --> 00:05:02.190
able to find an inverse.

00:05:02.190 --> 00:05:04.370
And when I defined determinant
in this way.

00:05:04.370 --> 00:05:06.590
If the determinant does not
equal 0, you will be able to

00:05:06.590 --> 00:05:07.210
find an inverse.

00:05:07.210 --> 00:05:09.200
So that's where this
came from.

00:05:09.200 --> 00:05:10.820
And I haven't shown
you that yet.

00:05:10.820 --> 00:05:12.540
And I might not show you
because it's super

00:05:12.540 --> 00:05:13.230
computational.

00:05:13.230 --> 00:05:14.580
It'll take a long time.

00:05:14.580 --> 00:05:16.260
It'll be very hairy and I'll
make careless mistakes.

00:05:16.260 --> 00:05:19.220
But the motivation comes from
the exact same place as the 2

00:05:19.220 --> 00:05:20.430
by 2 version.

00:05:20.430 --> 00:05:22.660
But I think what you probably
want to see right now is at

00:05:22.660 --> 00:05:26.050
least just this thing applied
to an actual matrix, because

00:05:26.050 --> 00:05:27.480
this looks all abstract
right now.

00:05:27.480 --> 00:05:29.790
But if we do it with an actual
matrix, you'll actually see

00:05:29.790 --> 00:05:31.490
it's not too bad.

00:05:31.490 --> 00:05:35.430
So let's leave the definition up
there, and let's say that I

00:05:35.430 --> 00:05:52.820
have the matrix 1, 2, 4, 2, 2,
minus 1, 3, and 4, 0, 1.

00:05:52.820 --> 00:05:56.070
So by our definition of a
determinant, the determinant

00:05:56.070 --> 00:06:00.410
of this guy right here-- so
let's say I call that matrix

00:06:00.410 --> 00:06:02.230
C-- C is equal to that.

00:06:02.230 --> 00:06:05.140
So if we want to figure out
the determinant of C, the

00:06:05.140 --> 00:06:09.690
determinant of C is equal to--
I take this guy right here,

00:06:09.690 --> 00:06:13.260
let me take that 1-- times the
determinant of-- let's just

00:06:13.260 --> 00:06:15.180
call it the submatrix,
right here.

00:06:15.180 --> 00:06:24.990
So we have a minus 1, we
have a 3, we have a 0,

00:06:24.990 --> 00:06:26.570
and we have a 1.

00:06:26.570 --> 00:06:27.520
Just like that.

00:06:27.520 --> 00:06:29.010
Notice, I got rid of
this guy's column

00:06:29.010 --> 00:06:29.690
and this guy's row.

00:06:29.690 --> 00:06:31.865
And I was just left with
minus 1, 3, 0, 1.

00:06:34.760 --> 00:06:38.460
Next, I take this guy.

00:06:38.460 --> 00:06:39.930
And this is the trick.

00:06:39.930 --> 00:06:41.160
You have to alternate signs.

00:06:41.160 --> 00:06:43.550
If you start with a positive
here, this next one's going to

00:06:43.550 --> 00:06:44.690
be a minus.

00:06:44.690 --> 00:06:49.710
So you're going to have minus 2
times the submatrix-- we can

00:06:49.710 --> 00:06:51.250
call it-- if we get rid
of this guy's column

00:06:51.250 --> 00:06:52.000
and this guy's row.

00:06:52.000 --> 00:06:54.630
So 2, 3, 4, 1.

00:06:59.050 --> 00:07:00.970
I just blanked this out.

00:07:00.970 --> 00:07:03.310
If I could videotape my finger,
I would cover my

00:07:03.310 --> 00:07:05.700
finger over this column right
here and over that row, and

00:07:05.700 --> 00:07:08.640
you'd just see a 2,
a 3, a 4, and a 1.

00:07:08.640 --> 00:07:10.280
And that's what I
put right there.

00:07:10.280 --> 00:07:14.720
And then finally, we went
plus, minus, plus.

00:07:14.720 --> 00:07:18.810
So finally, we'll have plus 4
times the determinant of the

00:07:18.810 --> 00:07:21.200
submatrix, if you get rid of
that row in that column.

00:07:21.200 --> 00:07:23.260
So 2, minus 1, 4, 0.

00:07:28.790 --> 00:07:30.660
Now, these are pretty
straightforward.

00:07:30.660 --> 00:07:32.660
These are not too
bad to compute.

00:07:32.660 --> 00:07:33.350
Let's actually do it.

00:07:33.350 --> 00:07:37.030
So this is going to be equal
to 1 times what?

00:07:37.030 --> 00:07:38.460
Minus 1 times 1.

00:07:38.460 --> 00:07:39.130
Let me just write it out.

00:07:39.130 --> 00:07:44.240
Minus 1 times 1, minus
0 times 3.

00:07:44.240 --> 00:07:46.330
This just comes from the
definition of a 2 by 2

00:07:46.330 --> 00:07:46.820
determinant.

00:07:46.820 --> 00:07:48.360
We've already defined that.

00:07:48.360 --> 00:07:55.300
And then we're going to have
a minus 2 times 2 times 1,

00:07:55.300 --> 00:07:58.260
minus 4 times 3.

00:07:58.260 --> 00:08:03.120
And then finally, we're going to
have a plus 4 times 2 times

00:08:03.120 --> 00:08:10.165
0 minus minus 1 times 4.

00:08:13.250 --> 00:08:15.410
I wrote it all out
so you can see.

00:08:15.410 --> 00:08:18.320
This thing right here is just
this thing right here.

00:08:18.320 --> 00:08:19.560
And then you have
the 4 out front.

00:08:19.560 --> 00:08:21.910
This thing right here was just
this thing right here.

00:08:21.910 --> 00:08:25.380
So it's the determinant of the
2 by 2 submatrix for each of

00:08:25.380 --> 00:08:26.350
these guys.

00:08:26.350 --> 00:08:30.910
And if we compute this, this is
equal to-- minus 1 times 1

00:08:30.910 --> 00:08:32.230
is minus 1.

00:08:32.230 --> 00:08:34.630
Minus 0, that's 0.

00:08:34.630 --> 00:08:38.440
So this is a minus 1 times
1, so that's a minus 1.

00:08:38.440 --> 00:08:41.950
And then we get-- what
is this equal to?

00:08:41.950 --> 00:08:43.580
This right here is 12.

00:08:43.580 --> 00:08:47.040
So you get 2 minus 12.

00:08:47.040 --> 00:08:47.530
Right?

00:08:47.530 --> 00:08:50.280
You get 2 times 1
minus 4 times 3.

00:08:50.280 --> 00:08:52.200
So it's minus 10.

00:08:52.200 --> 00:08:54.300
So that is equal to minus 10.

00:08:54.300 --> 00:08:58.000
And then you have a minus
10 times a minus 2.

00:08:58.000 --> 00:09:01.400
So that becomes a
plus 20, right?

00:09:01.400 --> 00:09:03.450
Minus 2 times minus 10.

00:09:03.450 --> 00:09:06.340
And then finally, in the green,
we have 2 times 0,

00:09:06.340 --> 00:09:07.770
that's just a 0.

00:09:07.770 --> 00:09:11.340
And then you have minus 1 times
4, which is minus 4.

00:09:11.340 --> 00:09:14.800
Then you have a minus sign
here, so it's plus 4.

00:09:14.800 --> 00:09:16.630
So this all becomes a plus 4.

00:09:16.630 --> 00:09:21.290
Plus 4 times 4 is
16, so plus 16.

00:09:21.290 --> 00:09:23.320
And what do we get when
we add this up?

00:09:23.320 --> 00:09:29.830
We get 20 plus 16 minus 1.

00:09:29.830 --> 00:09:33.430
It is equal to 35.

00:09:33.430 --> 00:09:34.030
We're done.

00:09:34.030 --> 00:09:38.570
We found the determinant
of our 3 by 3 matrix.

00:09:38.570 --> 00:09:41.380
Not too bad.

00:09:41.380 --> 00:09:47.450
Right there, so that is equal
to the determinant of C.

00:09:47.450 --> 00:09:50.040
So the fact that this isn't
0 tells you that C is

00:09:50.040 --> 00:09:51.290
invertible.

00:09:55.710 --> 00:09:58.950
In the next video, we'll try
to extend this to n by n

00:09:58.950 --> 00:10:00.200
square matrices.


Introduction to scientific notation

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:02.660
I don't think it's
any secret that if one

00:00:02.660 --> 00:00:04.285
were to do any kind
of science, they're

00:00:04.285 --> 00:00:06.035
going to be dealing
with a lot of numbers.

00:00:06.035 --> 00:00:08.340
It doesn't matter whether
you do biology, or chemistry,

00:00:08.340 --> 00:00:10.490
or physics, numbers
are involved.

00:00:10.490 --> 00:00:14.420
And in many cases, the
numbers are very large.

00:00:14.420 --> 00:00:19.760
They are very,
very large numbers.

00:00:19.760 --> 00:00:22.036
Very large numbers.

00:00:22.036 --> 00:00:26.220
Or, they're very, small,
very small numbers.

00:00:26.220 --> 00:00:29.000
Very small numbers.

00:00:29.000 --> 00:00:30.990
You could imagine some
very large numbers.

00:00:30.990 --> 00:00:32.770
If I were to ask
you, how many atoms

00:00:32.770 --> 00:00:33.990
are there in the human body?

00:00:33.990 --> 00:00:36.400
Or how cells are
in the human body?

00:00:36.400 --> 00:00:39.180
Or the mass of the
Earth, in kilograms,

00:00:39.180 --> 00:00:41.440
those are very large numbers.

00:00:41.440 --> 00:00:45.010
If I were to ask you
the mass of an electron,

00:00:45.010 --> 00:00:47.777
that would be a very,
very small number.

00:00:47.777 --> 00:00:50.360
So any kind of science, you're
going to be dealing with these.

00:00:50.360 --> 00:00:51.670
And just as an example,
let me show you

00:00:51.670 --> 00:00:53.794
one of the most common
numbers you're going to see,

00:00:53.794 --> 00:00:54.870
in especially chemistry.

00:00:54.870 --> 00:00:57.450
It's called Avogadro's number.

00:00:57.450 --> 00:01:00.840
Avogadro's number.

00:01:00.840 --> 00:01:04.110
And if I were write it in just
the standard way of writing

00:01:04.110 --> 00:01:06.090
a number, it would
literally be written

00:01:06.090 --> 00:01:08.380
as-- do it in a new color.

00:01:08.380 --> 00:01:12.880
It would be 6022-- and
then another 20 zeroes.

00:01:12.880 --> 00:01:21.222
1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17,

00:01:21.222 --> 00:01:23.290
18, 19, 20.

00:01:23.290 --> 00:01:25.320
And even I were to throw
some commas in here,

00:01:25.320 --> 00:01:27.111
it's not going to really
help the situation

00:01:27.111 --> 00:01:29.260
to make it more readable.

00:01:29.260 --> 00:01:31.990
Let me throw some
commas in here.

00:01:31.990 --> 00:01:33.860
This is still a huge number.

00:01:33.860 --> 00:01:35.940
If I have to write this
on a piece of paper

00:01:35.940 --> 00:01:38.582
or if I were to publish some
paper on using Avogadro's

00:01:38.582 --> 00:01:40.540
number, it would take me
forever to write this.

00:01:40.540 --> 00:01:43.400
And even more, it's hard to
tell if I forgot to write a zero

00:01:43.400 --> 00:01:46.420
or if I maybe wrote
too many zeroes.

00:01:46.420 --> 00:01:47.840
So there's a problem here.

00:01:47.840 --> 00:01:50.220
Is there a better
way to write this?

00:01:50.220 --> 00:01:55.680
So is there a better
way to write this

00:01:55.680 --> 00:01:57.490
than to write it
all out like this?

00:01:57.490 --> 00:02:00.220
To write literally the 6
followed by the 23 digits,

00:02:00.220 --> 00:02:05.190
or the 6022 followed
by the 20 zeroes there?

00:02:05.190 --> 00:02:07.122
And to answer that
question-- and in case

00:02:07.122 --> 00:02:08.830
you're curious,
Avogadro's number, if you

00:02:08.830 --> 00:02:11.230
had 12 grams of
carbon, especially

00:02:11.230 --> 00:02:14.312
12 grams of carbon-12,
this is how many atoms

00:02:14.312 --> 00:02:15.270
you would have in that.

00:02:15.270 --> 00:02:18.110
And just so you know, 12 grams
is like a 50th of a pound.

00:02:18.110 --> 00:02:20.110
So that just gives you
an idea of how many atoms

00:02:20.110 --> 00:02:22.330
are laying around at
any point in time.

00:02:22.330 --> 00:02:23.864
This is a huge number.

00:02:23.864 --> 00:02:26.030
The point of here isn't to
teach you some chemistry.

00:02:26.030 --> 00:02:28.360
The point of here is to
talk about an easier way

00:02:28.360 --> 00:02:29.470
to write this.

00:02:29.470 --> 00:02:33.960
And the easier way to write this
we call scientific notation.

00:02:33.960 --> 00:02:40.820
Scientific notation.

00:02:40.820 --> 00:02:42.517
And take my word
for it, although it

00:02:42.517 --> 00:02:44.600
might be a little unnatural
for you at this video.

00:02:44.600 --> 00:02:47.360
It really is an easier
way to write things

00:02:47.360 --> 00:02:49.170
like things like that.

00:02:49.170 --> 00:02:51.020
Before I show you
how to do it, let

00:02:51.020 --> 00:02:53.650
me show you the
underlying theory

00:02:53.650 --> 00:02:55.700
behind scientific notation.

00:02:55.700 --> 00:02:59.470
If I were to tell you,
what is 10 to the 0 power?

00:02:59.470 --> 00:03:01.070
We know that's equal to 1.

00:03:01.070 --> 00:03:03.040
What is 10 to the 1 power?

00:03:03.040 --> 00:03:04.380
That's equal to 10.

00:03:04.380 --> 00:03:06.220
What's 10 squared?

00:03:06.220 --> 00:03:07.440
That's 10 times 10.

00:03:07.440 --> 00:03:09.130
That's 100.

00:03:09.130 --> 00:03:11.270
What is 10 to the third?

00:03:11.270 --> 00:03:14.750
10 to the third is
10 times 10 times 10,

00:03:14.750 --> 00:03:16.940
which is equal to 1,000.

00:03:16.940 --> 00:03:18.950
I think you see a
general pattern here.

00:03:18.950 --> 00:03:21.810
10 to the 0 has no 0's.

00:03:21.810 --> 00:03:23.900
No 0's in it.

00:03:23.900 --> 00:03:27.800
10 to the 1 has one 0.

00:03:27.800 --> 00:03:32.330
10 to the second power-- I was
going to say the two-th power.

00:03:32.330 --> 00:03:36.360
10 to the second
power has two 0's.

00:03:36.360 --> 00:03:39.309
Finally, 10 to the
third has three 0's.

00:03:39.309 --> 00:03:40.850
Don't want to beat
a dead horse here,

00:03:40.850 --> 00:03:42.058
but I think you get the idea.

00:03:42.058 --> 00:03:44.750
Three 0's.

00:03:44.750 --> 00:03:48.790
If I were to do 10
to the 100th power,

00:03:48.790 --> 00:03:50.341
what would that look like?

00:03:50.341 --> 00:03:52.090
I don't feel like
writing it all out here,

00:03:52.090 --> 00:03:55.970
but it would be 1 followed
by-- you could guess it--

00:03:55.970 --> 00:03:56.960
a hundred 0's.

00:03:56.960 --> 00:03:59.190
So it would just
be a bunch of 0's.

00:03:59.190 --> 00:04:02.060
And if we were to count
up all of those 0's, you

00:04:02.060 --> 00:04:05.170
would have one hundred
0's right there.

00:04:05.170 --> 00:04:08.580
And actually, this might be
interesting, just as an aside.

00:04:08.580 --> 00:04:11.810
You may or may not know
what this number is called.

00:04:11.810 --> 00:04:16.210
This is called a googol.

00:04:16.210 --> 00:04:17.660
A googol.

00:04:17.660 --> 00:04:21.646
In the early '90s if someone
said, hey, that's a googol,

00:04:21.646 --> 00:04:23.520
you wouldn't have thought
of a search engine.

00:04:23.520 --> 00:04:25.530
You would have thought
of the number 10

00:04:25.530 --> 00:04:30.630
to the 100th power,
which is a huge number.

00:04:30.630 --> 00:04:33.590
It's more than the
number of atoms,

00:04:33.590 --> 00:04:37.590
or the estimated number of
atoms, in the known universe.

00:04:37.590 --> 00:04:38.940
In the known universe.

00:04:38.940 --> 00:04:41.260
It raises the question of
what else is there out there.

00:04:41.260 --> 00:04:45.500
But I was reading up on
this not too long ago.

00:04:45.500 --> 00:04:47.820
And if I remember correctly,
the known universe

00:04:47.820 --> 00:04:53.381
has the order of 10 to the
79th to 10 to the 81 atoms.

00:04:53.381 --> 00:04:54.630
And this is, of course, rough.

00:04:54.630 --> 00:04:55.680
No one can really count this.

00:04:55.680 --> 00:04:57.090
People are just kind
of estimating it.

00:04:57.090 --> 00:04:58.548
Or even better,
guesstimating this.

00:04:58.548 --> 00:05:00.044
But this is a huge number.

00:05:00.044 --> 00:05:01.710
What may be even more
interesting to you

00:05:01.710 --> 00:05:05.540
is this number was the
motivation behind the naming

00:05:05.540 --> 00:05:07.685
a very popular search
engine-- Google.

00:05:11.080 --> 00:05:13.160
Google is essentially
just a misspelling

00:05:13.160 --> 00:05:15.399
of the word "googol"
with the O-L.

00:05:15.399 --> 00:05:17.190
And I don't know why
they called it Google.

00:05:17.190 --> 00:05:18.790
Maybe they got the domain name.

00:05:18.790 --> 00:05:22.070
Maybe they want to hold
this much information.

00:05:22.070 --> 00:05:24.000
Maybe that many
bytes of information.

00:05:24.000 --> 00:05:25.700
Or, it's just a cool word.

00:05:25.700 --> 00:05:28.450
Whatever it is-- maybe it was
the founder's favorite number.

00:05:28.450 --> 00:05:29.700
But it's a cool thing to know.

00:05:29.700 --> 00:05:31.320
But anyway, I'm digressing.

00:05:31.320 --> 00:05:32.060
This is a googol.

00:05:32.060 --> 00:05:33.500
It's just 1 with a hundred 0's.

00:05:33.500 --> 00:05:37.060
But I could equivalently have
just written that as 10 to 100,

00:05:37.060 --> 00:05:40.200
which is clearly an easier way.

00:05:40.200 --> 00:05:42.160
This is an easier
way to write this.

00:05:42.160 --> 00:05:44.094
This is easier.

00:05:44.094 --> 00:05:46.260
In fact, this is so hard
to write that I didn't even

00:05:46.260 --> 00:05:47.190
take the trouble to write it.

00:05:47.190 --> 00:05:48.481
It would have taken me forever.

00:05:48.481 --> 00:05:50.210
This was just twenty
0's right here.

00:05:50.210 --> 00:05:53.320
A hundred 0's I would
have filled up this screen

00:05:53.320 --> 00:05:54.610
and you have found it boring.

00:05:54.610 --> 00:05:55.693
So I didn't even write it.

00:05:55.693 --> 00:05:57.560
So clearly, this
is easier to write.

00:06:01.190 --> 00:06:03.710
This is just good
for powers of 10.

00:06:03.710 --> 00:06:05.800
But how can we
write something that

00:06:05.800 --> 00:06:07.490
isn't a direct power of 10?

00:06:07.490 --> 00:06:10.400
How can we use the power
of this simplicity?

00:06:10.400 --> 00:06:13.510
How can we use the power
of the simplicity somehow?

00:06:13.510 --> 00:06:18.140
And to do that, you just
need to make the realization.

00:06:18.140 --> 00:06:20.610
This number, we
can write it as--

00:06:20.610 --> 00:06:22.220
so this has how
many digits in it?

00:06:22.220 --> 00:06:26.430
It has 1, 2, 3, and
then twenty 0's.

00:06:26.430 --> 00:06:30.880
So it has 23 digits after the 6.

00:06:30.880 --> 00:06:33.430
23 digits after the 6.

00:06:33.430 --> 00:06:37.150
So what happens if I use this--
if I try to get close to it

00:06:37.150 --> 00:06:39.020
with a power of 10?

00:06:39.020 --> 00:06:41.420
So what if I were
to say 10 to the 23?

00:06:41.420 --> 00:06:43.170
Do it in this magenta.

00:06:43.170 --> 00:06:44.775
10 to the 23rd power.

00:06:44.775 --> 00:06:45.650
That's equal to what?

00:06:45.650 --> 00:06:48.450
That equals 1 with 23 0's.

00:06:48.450 --> 00:06:57.142
So 1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17,

00:06:57.142 --> 00:06:59.480
18, 19, 20, 21, 22, 23.

00:06:59.480 --> 00:07:02.280
You get the idea,
that's 10 to the 23rd.

00:07:02.280 --> 00:07:04.920
Now, can we somehow
write this guy

00:07:04.920 --> 00:07:08.080
as some multiple of this guy?

00:07:08.080 --> 00:07:08.690
Well, we can.

00:07:08.690 --> 00:07:13.931
Because if we multiplied
this guy by 6--

00:07:13.931 --> 00:07:18.410
if we multiply 6 times 10
to the 23rd, what do we get?

00:07:18.410 --> 00:07:21.000
Well, we're just going to have
a 6 with twenty three 0's.

00:07:21.000 --> 00:07:24.020
We're going to have a 6, and
then you're going to have

00:07:24.020 --> 00:07:25.090
twenty three 0's.

00:07:25.090 --> 00:07:25.840
Let me write that.

00:07:25.840 --> 00:07:27.947
You're going to have
twenty three 0's like that.

00:07:27.947 --> 00:07:29.780
Because all I did, if
you take 6 times this.

00:07:29.780 --> 00:07:30.821
You know how to multiply.

00:07:30.821 --> 00:07:32.240
You'd have the 6 times this 1.

00:07:32.240 --> 00:07:32.960
You'd get a 6.

00:07:32.960 --> 00:07:34.920
And then all the 6 times
the 0's will all be 0.

00:07:34.920 --> 00:07:37.650
So you'll have 6 followed
by twenty three 0's.

00:07:37.650 --> 00:07:38.960
So that's pretty useful.

00:07:38.960 --> 00:07:41.190
But still, we're not getting
quite to this number.

00:07:41.190 --> 00:07:43.600
I mean, this had
some 2's in there.

00:07:43.600 --> 00:07:45.350
So how could we do it
a little bit better?

00:07:45.350 --> 00:07:46.975
Well, what if we
wrote it as a decimal?

00:07:46.975 --> 00:07:51.240
This number right here is
identical to this number

00:07:51.240 --> 00:07:52.920
if these 2's were 0's.

00:07:52.920 --> 00:07:55.690
But if we want to put those
2's there, what can we do?

00:07:55.690 --> 00:07:57.600
We could put some decimals here.

00:07:57.600 --> 00:08:02.570
We could say that this is the
same thing as 6.022 times 10

00:08:02.570 --> 00:08:03.770
to the 23rd.

00:08:03.770 --> 00:08:06.860
And now, this number is
identical to this number,

00:08:06.860 --> 00:08:08.920
but it's a much easier
way to write it.

00:08:08.920 --> 00:08:10.542
And you could verify
it, if you like.

00:08:10.542 --> 00:08:11.750
It will take you a long time.

00:08:11.750 --> 00:08:13.870
Maybe we should do it with
a smaller number first.

00:08:13.870 --> 00:08:18.080
But if you multiply 6.022
times 10 to the 23rd,

00:08:18.080 --> 00:08:20.870
and you write it all out,
you will get this number

00:08:20.870 --> 00:08:21.390
right there.

00:08:21.390 --> 00:08:23.403
You will get Avogadro's number.

00:08:26.970 --> 00:08:29.516
Avogadro's number.

00:08:29.516 --> 00:08:31.890
And although this is complicated
or it looks a little bit

00:08:31.890 --> 00:08:34.350
unintuitive to you at first,
this was just a number

00:08:34.350 --> 00:08:35.220
written out.

00:08:35.220 --> 00:08:37.542
This has a multiplication
and then a 10 to a power.

00:08:37.542 --> 00:08:39.250
You might say, hey,
that's not so simple.

00:08:39.250 --> 00:08:40.390
But it really is.

00:08:40.390 --> 00:08:43.690
Because you immediately
know how many 0's there are.

00:08:43.690 --> 00:08:46.755
And it's obviously
a much shorter way

00:08:46.755 --> 00:08:47.630
to write this number.

00:08:47.630 --> 00:08:49.110
Let's do a couple of more.

00:08:49.110 --> 00:08:51.230
I started with Avogadro's
number because it really

00:08:51.230 --> 00:08:54.759
shows you the need for
a scientific notation.

00:08:54.759 --> 00:08:56.550
So you don't have to
write things like that

00:08:56.550 --> 00:08:57.530
over and over again.

00:08:57.530 --> 00:08:59.113
So let's do a couple
of other numbers.

00:08:59.113 --> 00:09:01.940
And we'll just write them
in scientific notation.

00:09:01.940 --> 00:09:09.130
So let's say I have
the number 7,345.

00:09:09.130 --> 00:09:11.910
And I wanted to write it
in scientific notation.

00:09:11.910 --> 00:09:18.030
So I guess the best way to
think about it is, it's 7,345.

00:09:18.030 --> 00:09:20.470
So how can I
represent a thousand?

00:09:20.470 --> 00:09:23.530
Well, I wrote it over here,
10 to the third is 1,000.

00:09:23.530 --> 00:09:28.480
So we know that 10 to the
third is equal to 1,000.

00:09:28.480 --> 00:09:30.870
So that's essentially
the largest power of 10

00:09:30.870 --> 00:09:32.070
that I can fit into this.

00:09:32.070 --> 00:09:33.600
This is seven 1,000's.

00:09:33.600 --> 00:09:36.880
So if this is seven 1,000's,
and then it's 0.3 1,00's, then

00:09:36.880 --> 00:09:39.790
it's 0.4 1,000's-- I don't
know if that helps you,

00:09:39.790 --> 00:09:45.950
we can write this as 7.345
times 10 to the third

00:09:45.950 --> 00:09:49.460
because it's going to be seven
1,000's plus 0.3 1,000's.

00:09:49.460 --> 00:09:51.640
What's 0.3 times 1,000?

00:09:51.640 --> 00:09:53.270
0.3 times 1,000 is 300.

00:09:53.270 --> 00:09:55.550
What's 0.04 times 1,000?

00:09:55.550 --> 00:09:56.150
That's 40.

00:09:56.150 --> 00:09:58.420
What's 0.05 times 1,000?

00:09:58.420 --> 00:09:59.170
That's a 5.

00:09:59.170 --> 00:10:05.870
So 7.345 times 1,000
is equal to 7,345.

00:10:05.870 --> 00:10:09.450
Let me multiply it out
just to make it clear.

00:10:09.450 --> 00:10:15.190
So if I took 7.345 times 1,000.

00:10:15.190 --> 00:10:17.420
The way I do it is
I ignore the 0's.

00:10:17.420 --> 00:10:21.130
I essentially multiply 1
times that guy up there.

00:10:21.130 --> 00:10:24.890
So I get 7, 3, 4, 5.

00:10:24.890 --> 00:10:28.090
Then I had three 0's here,
so I put those on the end.

00:10:28.090 --> 00:10:30.000
And then I have
three decimal places.

00:10:30.000 --> 00:10:30.890
1, 2, 3.

00:10:30.890 --> 00:10:31.890
So 1, 2, 3.

00:10:31.890 --> 00:10:33.470
Put the decimal right there.

00:10:33.470 --> 00:10:40.270
And there you have it, 7.345
times 1,000 is indeed 7,345.

00:10:40.270 --> 00:10:41.354
Let's do a couple of them.

00:10:41.354 --> 00:10:42.978
Let's say we wanted
to write the number

00:10:42.978 --> 00:10:44.360
6 in scientific notation.

00:10:44.360 --> 00:10:46.818
Obviously, there's no need to
write in scientific notation.

00:10:46.818 --> 00:10:47.950
But how would you do it?

00:10:47.950 --> 00:10:51.390
Well, what's the largest
power of 10 that fits into 6?

00:10:51.390 --> 00:10:54.540
Well, the largest power of 10
that fits into 6 is just 1.

00:10:54.540 --> 00:10:59.180
So we could write it as
something times 10 to the 0.

00:10:59.180 --> 00:11:00.670
This is just 1, right?

00:11:00.670 --> 00:11:01.910
That's just 1.

00:11:01.910 --> 00:11:03.640
So 6 is what times 1?

00:11:03.640 --> 00:11:04.940
Well, it's just 6.

00:11:04.940 --> 00:11:07.180
So 6 is equal to 6
times 10 to the 0.

00:11:07.180 --> 00:11:10.360
You wouldn't actually
have to write it this way.

00:11:10.360 --> 00:11:12.023
This is much simpler,
but it shows you

00:11:12.023 --> 00:11:17.270
that you really can express any
number in scientific notation.

00:11:17.270 --> 00:11:20.370
Now, what if we wanted to
represent something like this?

00:11:20.370 --> 00:11:22.850
I had started off the
video saying in science

00:11:22.850 --> 00:11:26.630
you deal with very large
and very small numbers.

00:11:26.630 --> 00:11:29.940
So let's say you had the
number-- do it in this color.

00:11:29.940 --> 00:11:33.280
And you had 1, 2, 3, 4.

00:11:33.280 --> 00:11:35.660
And then, let's say five 0's.

00:11:35.660 --> 00:11:38.760
And then you have
followed by a 7.

00:11:38.760 --> 00:11:41.210
Well, once again, this is not
an easy number to deal with.

00:11:41.210 --> 00:11:44.050
But how can we deal with
it as a power of 10?

00:11:44.050 --> 00:11:45.450
As a power of 10?

00:11:45.450 --> 00:11:48.605
So what's the largest power of
10 that fits into this number,

00:11:48.605 --> 00:11:50.850
that this number
is divisible by?

00:11:50.850 --> 00:11:51.850
So let's think about it.

00:11:51.850 --> 00:11:54.870
All the powers of 10 we did
before were going to positive

00:11:54.870 --> 00:11:57.790
or going to-- well, yeah,
positive powers of 10.

00:11:57.790 --> 00:11:59.890
We could also do
negative powers of 10.

00:11:59.890 --> 00:12:01.730
We know that 10 to the 0 is 1.

00:12:01.730 --> 00:12:02.940
Let's start there.

00:12:02.940 --> 00:12:08.985
10 to the minus 1 is equal to
1/10, which is equal to 0.1.

00:12:08.985 --> 00:12:09.860
Let me switch colors.

00:12:09.860 --> 00:12:11.030
I'll do pink.

00:12:11.030 --> 00:12:15.430
10 to the minus 2 is equal
to 1 over 10 squared, which

00:12:15.430 --> 00:12:20.610
is equal to 1/100,
which is equal to 0.01.

00:12:20.610 --> 00:12:23.639
And you I think you get the
idea that the--, well, let

00:12:23.639 --> 00:12:25.680
me just do one more so
that you can get the idea.

00:12:25.680 --> 00:12:27.266
10 to the minus 3.

00:12:27.266 --> 00:12:31.880
10 to the minus 3 is equal
to 1 over 10 to the third,

00:12:31.880 --> 00:12:37.300
which is equal to 1/1,000,
which is equal to the 0.001.

00:12:37.300 --> 00:12:39.640
So the general pattern
here is 10 to the whatever

00:12:39.640 --> 00:12:42.780
negative power is however
many places you're

00:12:42.780 --> 00:12:44.430
going to have behind
the decimal point.

00:12:44.430 --> 00:12:46.440
So here, it's not
the number of 0's.

00:12:46.440 --> 00:12:48.390
In here, 10 to the
minus 3, you only

00:12:48.390 --> 00:12:50.390
have two 0's but you
have three places

00:12:50.390 --> 00:12:52.290
behind the decimal point.

00:12:52.290 --> 00:12:56.150
So what is the largest power
of 10 that goes into this?

00:12:56.150 --> 00:13:00.032
Well, how many places behind
the decimal point do I have?

00:13:00.032 --> 00:13:04.460
I have 1, 2, 3, 4, 5, 6.

00:13:04.460 --> 00:13:09.100
So 10 to the minus 6 is
going to be equal to 0.--

00:13:09.100 --> 00:13:10.619
and we're going
to have six places

00:13:10.619 --> 00:13:11.660
behind the decimal point.

00:13:11.660 --> 00:13:14.580
And the last place
is going to be a 1.

00:13:14.580 --> 00:13:17.220
So you're going to
have Five 0's and a 1.

00:13:17.220 --> 00:13:19.790
That's 10 to the minus 6.

00:13:19.790 --> 00:13:23.290
Now, this number right here
is 7 times this number.

00:13:23.290 --> 00:13:29.120
If we multiply this times
7, we get 7 times 1.

00:13:29.120 --> 00:13:31.959
And then we have 1,
2, 3, 4, 5, 6 numbers

00:13:31.959 --> 00:13:33.000
behind the decimal point.

00:13:33.000 --> 00:13:36.710
So 1, 2, 3, 4, 5, 6.

00:13:36.710 --> 00:13:40.950
So this number times 7 is
clearly equal to the number

00:13:40.950 --> 00:13:42.490
that we started off with.

00:13:42.490 --> 00:13:46.280
So we can rewrite this number.

00:13:46.280 --> 00:13:48.570
Instead of writing
this number every time,

00:13:48.570 --> 00:13:51.410
we can write it as being
equal to this number.

00:13:51.410 --> 00:13:53.090
Or, we could write it as 7.

00:13:53.090 --> 00:13:55.604
This is equal to 7
times this number.

00:13:55.604 --> 00:13:57.520
But this number is no
better than that number.

00:13:57.520 --> 00:14:00.320
But this number is the same
thing as 10 to the minus 6.

00:14:00.320 --> 00:14:03.640
7 times 10 to the minus 6.

00:14:03.640 --> 00:14:07.170
So now you can imagine
numbers like-- imagine

00:14:07.170 --> 00:14:10.570
the number-- what
if we had a 7-- let

00:14:10.570 --> 00:14:12.700
me think of it this way.

00:14:12.700 --> 00:14:16.630
Let's say we had
a 7, 3 over there.

00:14:16.630 --> 00:14:17.830
So what would we do?

00:14:17.830 --> 00:14:19.850
Well, we'd want to
go to the first digit

00:14:19.850 --> 00:14:23.730
right here because this is
kind of the largest power of 10

00:14:23.730 --> 00:14:25.714
that could go into
this thing right here.

00:14:25.714 --> 00:14:27.380
So if we wanted to
represent that thing,

00:14:27.380 --> 00:14:30.330
let me do another decimal
that's like that one.

00:14:30.330 --> 00:14:35.990
So let's say I did 0.0000516
and I wanted to represent this

00:14:35.990 --> 00:14:38.010
in scientific notation.

00:14:38.010 --> 00:14:40.370
I'd go to the
first non-digit 0--

00:14:40.370 --> 00:14:45.970
the first non-zero digit, not
non-digit 0, which is there.

00:14:45.970 --> 00:14:47.970
And I'm like, OK, what's
the largest power of 10

00:14:47.970 --> 00:14:48.970
that will fit into that?

00:14:48.970 --> 00:14:52.770
So I'll go 1, 2, 3, 4, 5.

00:14:52.770 --> 00:14:57.230
So it's going to
be equal to 5.16.

00:14:57.230 --> 00:14:59.400
So I take 5 there,
then everything else

00:14:59.400 --> 00:15:01.180
is going to be behind
the decimal point.

00:15:01.180 --> 00:15:02.505
Times 10.

00:15:02.505 --> 00:15:04.630
So this is going to be the
largest power of 10 that

00:15:04.630 --> 00:15:07.500
fits into this first
non-zero number.

00:15:07.500 --> 00:15:09.740
So it's 1, 2, 3, 4, 5.

00:15:09.740 --> 00:15:12.790
So 10 to the minus 5 power.

00:15:12.790 --> 00:15:13.950
Let me do another example.

00:15:13.950 --> 00:15:17.260
So the point I wanted to make
is you just go to the first--

00:15:17.260 --> 00:15:19.930
if you're starting at the left,
the first non-zero number.

00:15:19.930 --> 00:15:22.040
That's what you get
your power from.

00:15:22.040 --> 00:15:23.700
That's where i got
my 10 to the minus 5

00:15:23.700 --> 00:15:25.740
because I counted 1, 2, 3, 4, 5.

00:15:25.740 --> 00:15:28.380
You got to count that number
just like we did over here.

00:15:28.380 --> 00:15:31.040
And then, everything else
will be behind the decimal.

00:15:31.040 --> 00:15:33.210
Let me do another example.

00:15:33.210 --> 00:15:35.370
Let's say I had 0.-- and
my wife always point out

00:15:35.370 --> 00:15:37.620
that I have to write a 0 in
front of my decimal points

00:15:37.620 --> 00:15:38.580
because she's a doctor.

00:15:38.580 --> 00:15:40.330
And if people don't
see the decimal point,

00:15:40.330 --> 00:15:42.420
someone might overdose
on some medication.

00:15:42.420 --> 00:15:52.350
So let's write it her
way, 0.0000000008192.

00:15:52.350 --> 00:15:54.800
Clearly, this is a super
cumbersome number to write.

00:15:54.800 --> 00:15:56.850
And you might forget
about a 0 or add too

00:15:56.850 --> 00:15:59.310
many 0's, which could
be costly if you're

00:15:59.310 --> 00:16:01.260
doing some important
scientific research.

00:16:01.260 --> 00:16:03.605
Or, maybe doing-- well, you
wouldn't prescribe medicine

00:16:03.605 --> 00:16:04.480
at this small a dose.

00:16:04.480 --> 00:16:06.563
Or maybe you would, I don't
want to get into that.

00:16:06.563 --> 00:16:09.200
But how would I write this
in scientific notation?

00:16:09.200 --> 00:16:12.220
So I start off with the
first non-zero number,

00:16:12.220 --> 00:16:13.690
if I'm starting from the left.

00:16:13.690 --> 00:16:16.550
So it's going to be 8.192.

00:16:16.550 --> 00:16:21.310
I just put a decimal and write
0.192 times-- times 10 to what?

00:16:21.310 --> 00:16:23.040
Well, I just count.

00:16:23.040 --> 00:16:28.600
Times 10 to the 1, 2,
3, 4, 5, 6, 7, 8, 9, 10.

00:16:28.600 --> 00:16:32.110
I have to include that
number, 10 to the minus 10.

00:16:32.110 --> 00:16:35.080
And I think you'll find
it reasonably satisfactory

00:16:35.080 --> 00:16:37.890
that this number
is easier to write

00:16:37.890 --> 00:16:40.030
than that number over there.

00:16:40.030 --> 00:16:42.180
Now, and this is
another powerful thing

00:16:42.180 --> 00:16:44.430
about scientific notation.

00:16:44.430 --> 00:16:45.934
Let's say I have
these two numbers

00:16:45.934 --> 00:16:47.100
and I want to multiply them.

00:16:47.100 --> 00:16:53.810
Let's say I want to multiply the
number 0.005 times the number

00:16:53.810 --> 00:16:58.540
0.0008.

00:16:58.540 --> 00:17:01.630
This is actually a fairly
straightforward one to do,

00:17:01.630 --> 00:17:04.155
but sometimes it can
get quite cumbersome.

00:17:04.155 --> 00:17:06.500
And especially if you're
dealing with twenty or thirty

00:17:06.500 --> 00:17:08.849
0's on either sides
of the decimal point.

00:17:08.849 --> 00:17:10.740
Put a 0 here to
make my wife happy.

00:17:10.740 --> 00:17:12.490
But when you do it in
scientific notation,

00:17:12.490 --> 00:17:14.069
it will actually simplify it.

00:17:14.069 --> 00:17:18.420
This guy can be rewritten
as 5 times 10 to the what?

00:17:18.420 --> 00:17:20.690
I have 1, 2, 3 spaces
behind the decimal.

00:17:20.690 --> 00:17:21.960
10 to the third.

00:17:21.960 --> 00:17:25.560
And then this is 8, so
this is times 8 times

00:17:25.560 --> 00:17:29.704
10 to the-- sorry, this is
5 times 10 to the minus 3.

00:17:29.704 --> 00:17:30.620
That's very important.

00:17:30.620 --> 00:17:33.380
5 times 10 to the 3
would have been 5,000.

00:17:33.380 --> 00:17:34.909
Be very careful about that.

00:17:34.909 --> 00:17:36.200
Now, what is this guy equal to?

00:17:36.200 --> 00:17:39.740
This is 1, 2, 3, 4 places
behind the decimal.

00:17:39.740 --> 00:17:44.567
So it's 8 times
10 to the minus 4.

00:17:44.567 --> 00:17:46.150
If we're multiplying
these two things,

00:17:46.150 --> 00:17:49.870
this is the same thing as
5 times 10 to the minus 3

00:17:49.870 --> 00:17:52.950
times 8 times 10 to the minus 4.

00:17:52.950 --> 00:17:55.200
There's nothing special about
the scientific notation.

00:17:55.200 --> 00:17:56.820
It literally means
what it's saying.

00:17:56.820 --> 00:17:59.070
So for multiplying, you
could write it out like this.

00:17:59.070 --> 00:18:01.610
And multiplication,
order doesn't matter.

00:18:01.610 --> 00:18:05.460
So I could rewrite
this as 5 times 8 times

00:18:05.460 --> 00:18:09.200
10 to the minus 3 times
10 to the minus 4.

00:18:09.200 --> 00:18:11.860
And then, what is 5 times 8?

00:18:11.860 --> 00:18:15.900
5 times 8 we know is 40.

00:18:15.900 --> 00:18:20.830
So it's 40 times 10 to the
minus 3 times 10 to the minus 4.

00:18:20.830 --> 00:18:22.330
And if you know
your exponent rules,

00:18:22.330 --> 00:18:24.620
you know that when you
multiply two numbers that

00:18:24.620 --> 00:18:27.400
have the same base, you can
just add their exponents.

00:18:27.400 --> 00:18:29.820
So you just add the
minus 3 and the minus 4.

00:18:29.820 --> 00:18:35.080
So it's equal to 40
times 10 to the minus 7.

00:18:35.080 --> 00:18:36.160
Let's do another example.

00:18:36.160 --> 00:18:39.580
Let's say we were to
multiply Avogadro's number.

00:18:39.580 --> 00:18:44.230
So we know that's 6.022
times 10 to the 23rd.

00:18:44.230 --> 00:18:47.740
Now, let's say we multiply that
times some really small number.

00:18:47.740 --> 00:18:57.900
So times, say, 7.23
times 10 to the minus 22.

00:18:57.900 --> 00:18:59.530
So this is some
really small number.

00:18:59.530 --> 00:19:02.071
You're going to have a decimal,
and then you're going to have

00:19:02.071 --> 00:19:02.980
twenty one 0's.

00:19:02.980 --> 00:19:05.180
Then you're going ti
have a 7 and a 2 and a 3.

00:19:05.180 --> 00:19:07.260
So this is a really
small number.

00:19:07.260 --> 00:19:09.990
But the multiplication, when you
do it in scientific notation,

00:19:09.990 --> 00:19:12.050
is actually fairly
straightforward.

00:19:12.050 --> 00:19:16.490
This is going to be equal to
6.0-- let me write it properly.

00:19:16.490 --> 00:19:24.920
6.022 times 10 to the 23rd times
7.23 times 10 to the minus 22.

00:19:24.920 --> 00:19:32.570
We can change the order, so
it's equal to 6.022 times 7.23.

00:19:32.570 --> 00:19:33.430
That's that part.

00:19:33.430 --> 00:19:35.400
So you can view it
as these first parts

00:19:35.400 --> 00:19:42.540
of our scientific notation
times 10 to the 23rd times 10

00:19:42.540 --> 00:19:44.680
to the minus 22.

00:19:44.680 --> 00:19:46.534
And now, this is--
you're going to do

00:19:46.534 --> 00:19:48.450
some little decimal
multiplication right here.

00:19:48.450 --> 00:19:52.600
It's going to be-- some
number-- 40 something, I think.

00:19:52.600 --> 00:19:54.140
I can't do this one in my head.

00:19:54.140 --> 00:19:56.040
But this part is pretty
easy to calculate.

00:19:56.040 --> 00:19:58.620
I'll just leave
this the way it is.

00:19:58.620 --> 00:20:00.990
But this part right
here, this will be times.

00:20:00.990 --> 00:20:02.820
10 to the 23rd times
10 to the minus 22.

00:20:02.820 --> 00:20:03.944
You just add the exponents.

00:20:03.944 --> 00:20:07.125
You get times 10
to the first power.

00:20:07.125 --> 00:20:09.250
And then this number,
whatever it's going to equal,

00:20:09.250 --> 00:20:10.708
I'll just leave it
out here since I

00:20:10.708 --> 00:20:12.140
don't have a calculator.

00:20:12.140 --> 00:20:13.370
0.23.

00:20:13.370 --> 00:20:15.490
Let's see, it will be 7.2.

00:20:15.490 --> 00:20:17.820
Let's see, 0.2 times--
it's like a fifth.

00:20:17.820 --> 00:20:19.400
It'll be like 41-something.

00:20:19.400 --> 00:20:25.170
So this is approximately
41 times 10 to the 1.

00:20:25.170 --> 00:20:27.020
Or, another way
is approximately--

00:20:27.020 --> 00:20:29.430
it's going to be 410-something.

00:20:29.430 --> 00:20:31.220
And to get it
right, you just have

00:20:31.220 --> 00:20:33.310
to actually perform
this multiplication.

00:20:33.310 --> 00:20:36.210
So hopefully you see that
scientific notation is, one,

00:20:36.210 --> 00:20:39.940
really useful for super large
and super small numbers.

00:20:39.940 --> 00:20:42.400
And not only is it more
useful to kind of understand

00:20:42.400 --> 00:20:43.950
the numbers and to
write the numbers,

00:20:43.950 --> 00:20:46.875
but it also simplifies
operating on the numbers.


Formula for 2x2 inverse

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.980
I've got a 2 by 2 matrix here.

00:00:02.980 --> 00:00:07.430
Let's say A is equal
to a, b, c, d.

00:00:07.430 --> 00:00:08.990
So I'm going to keep
it really general.

00:00:08.990 --> 00:00:11.570
So this is really any
2 by 2 matrix.

00:00:11.570 --> 00:00:15.070
What I want to do is use our
technique for finding an

00:00:15.070 --> 00:00:18.920
inverse of this matrix to
essentially find a formula for

00:00:18.920 --> 00:00:21.610
the inverse of a
2 by 2 matrix.

00:00:21.610 --> 00:00:25.170
So I want to essentially find a
inverse, and I want to do it

00:00:25.170 --> 00:00:28.390
just using a formula that it
just applies to this matrix

00:00:28.390 --> 00:00:28.960
right here.

00:00:28.960 --> 00:00:30.020
So how can I do that?

00:00:30.020 --> 00:00:31.300
Well, we know a technique.

00:00:31.300 --> 00:00:33.690
We just create an augmented
matrix.

00:00:33.690 --> 00:00:37.330
So let's just create an
augmented matrix right here.

00:00:37.330 --> 00:00:43.110
So we have a, b, c, d, and then
we augment it with the

00:00:43.110 --> 00:00:47.920
identity in R2, so 1, 0, 0, 1.

00:00:47.920 --> 00:00:50.410
And we know if we perform a
series of row operations on

00:00:50.410 --> 00:00:53.680
this augmented matrix to get the
left-hand side in reduced

00:00:53.680 --> 00:00:54.880
row echelon form.

00:00:54.880 --> 00:00:57.500
The right-hand side, if the
reduced row echelon form here

00:00:57.500 --> 00:01:00.790
gets to the identity, then the
right-hand side is going to be

00:01:00.790 --> 00:01:01.710
the inverse.

00:01:01.710 --> 00:01:04.780
So let's do it in this general
case, not dealing with

00:01:04.780 --> 00:01:06.310
particular numbers here.

00:01:06.310 --> 00:01:08.990
So the first thing I want to do,
or I would like to do, is

00:01:08.990 --> 00:01:11.050
I would like to zero
this guy out.

00:01:11.050 --> 00:01:13.060
What we want to do is we want
to zero that out, zero that

00:01:13.060 --> 00:01:15.860
out, and then these two terms
have to become equal to 1.

00:01:15.860 --> 00:01:20.810
So the best way to zero this
out, let's perform a little

00:01:20.810 --> 00:01:22.470
transformation here.

00:01:22.470 --> 00:01:29.790
So if I perform the
transformation on the columns,

00:01:29.790 --> 00:01:32.560
C1, so those are the entries
of a column-- this would be

00:01:32.560 --> 00:01:35.780
one column right here, that
would be another column right

00:01:35.780 --> 00:01:37.440
there, that's the third column,

00:01:37.440 --> 00:01:38.210
that's the fourth column.

00:01:38.210 --> 00:01:40.170
But the transformation I'm going
to perform on each of

00:01:40.170 --> 00:01:41.940
these columns, and we know this
is equivalent to a row

00:01:41.940 --> 00:01:44.800
operation, is going to be equal
to-- since I want to

00:01:44.800 --> 00:01:49.580
zero this one out, I'm going to
keep my first row the same,

00:01:49.580 --> 00:01:52.000
so it's going to be C1, and I'm
going to replace my second

00:01:52.000 --> 00:01:59.430
row with a times my second row
minus c times my first row.

00:01:59.430 --> 00:02:00.790
Now why am I doing that?

00:02:00.790 --> 00:02:04.240
Because a times c minus c times
a is going to be 0.

00:02:04.240 --> 00:02:05.820
So this guy's going to be 0.

00:02:05.820 --> 00:02:07.890
That's the row operation
I'm going to perform.

00:02:07.890 --> 00:02:10.259
And I'm doing this so we can
kind of keep track, account

00:02:10.259 --> 00:02:12.110
for what we're doing because
the algebra's going to get

00:02:12.110 --> 00:02:13.305
hairy in a little bit.

00:02:13.305 --> 00:02:15.920
So let me perform
this operation.

00:02:15.920 --> 00:02:18.920
So if I perform that operation
on our matrix, what do we

00:02:18.920 --> 00:02:19.950
have.

00:02:19.950 --> 00:02:22.320
So our first row's going
to be the same.

00:02:22.320 --> 00:02:24.010
Let me start with our second
row because that's a little

00:02:24.010 --> 00:02:25.570
bit more complicated.

00:02:25.570 --> 00:02:31.030
So I'm going to replace c with
a times c minus c times a.

00:02:31.030 --> 00:02:34.710
That's ac-- so let me put it
this way-- so that's going to

00:02:34.710 --> 00:02:36.930
be 0 right there.

00:02:36.930 --> 00:02:45.820
I'm going to replace d with d
times a or a times d minus c

00:02:45.820 --> 00:02:48.200
times C1 in this
column vector.

00:02:48.200 --> 00:02:50.360
So minus c times b.

00:02:50.360 --> 00:02:52.940
Let me write this as bc.

00:02:52.940 --> 00:02:55.220
And then let me augment it.

00:02:55.220 --> 00:02:58.830
And then this guy's going to be
a times 0, because he's C2

00:02:58.830 --> 00:03:00.480
minus c times C1.

00:03:00.480 --> 00:03:03.580
So it's going to be minus c.

00:03:03.580 --> 00:03:07.340
And then finally, this guy right
here is going to be a

00:03:07.340 --> 00:03:12.450
times 1-- a times this 1 right
here-- minus c times 0.

00:03:12.450 --> 00:03:14.870
So that's just going
to be an a.

00:03:14.870 --> 00:03:17.330
And then the first row is
pretty straightforward.

00:03:17.330 --> 00:03:19.980
We know that the first row or
the first entries in our

00:03:19.980 --> 00:03:21.820
column vectors just stay
the same through this

00:03:21.820 --> 00:03:23.450
transformation.

00:03:23.450 --> 00:03:27.790
So it's a, b, 1, 0.

00:03:27.790 --> 00:03:30.000
And just to make sure you're
clear what we're doing, when

00:03:30.000 --> 00:03:32.400
you perform this transformation
on this column

00:03:32.400 --> 00:03:35.100
vector right here, you
got this column

00:03:35.100 --> 00:03:36.550
vector right there.

00:03:36.550 --> 00:03:38.450
When you perform the
transformation on this column

00:03:38.450 --> 00:03:40.490
vector right there, you
get this column

00:03:40.490 --> 00:03:41.560
vector right there.

00:03:41.560 --> 00:03:44.020
Now I just want to make that
clear because I did all of the

00:03:44.020 --> 00:03:48.200
second entries of all of the
column vectors at once because

00:03:48.200 --> 00:03:50.080
we all were essentially
performing the same row

00:03:50.080 --> 00:03:52.800
operation, so that just helped
me simplify at least my

00:03:52.800 --> 00:03:53.570
thinking a little bit.

00:03:53.570 --> 00:03:55.460
Let me stay in this mode.

00:03:55.460 --> 00:03:58.180
So let's continue to get this
in reduced row echelon form.

00:03:58.180 --> 00:04:00.020
The next thing we want to do
is, let's make another

00:04:00.020 --> 00:04:02.020
transformation, we'll
call this T1.

00:04:02.020 --> 00:04:03.780
That was our first
transformation.

00:04:03.780 --> 00:04:05.180
Let's do another
transformation.

00:04:05.180 --> 00:04:08.350
T2, or another set of
row operations.

00:04:08.350 --> 00:04:13.490
So if I start with the column
vector C1, C2, what I want to

00:04:13.490 --> 00:04:16.790
do now is I want to keep my
second row the same and I want

00:04:16.790 --> 00:04:19.019
to zero out this character
right here.

00:04:19.019 --> 00:04:20.760
I want to zero him out.

00:04:20.760 --> 00:04:24.490
I know I'm going to keep my
second row the same, so C2 is

00:04:24.490 --> 00:04:27.680
just going to still be C2.

00:04:27.680 --> 00:04:30.470
But in order to zero this out,
what I can do is I can replace

00:04:30.470 --> 00:04:35.000
the first row with this scaling
factor times the first

00:04:35.000 --> 00:04:38.250
row minus this scaling factor
times the second row.

00:04:38.250 --> 00:04:44.970
So it'll be ad minus bc times
your first entry in your

00:04:44.970 --> 00:04:50.260
column vector minus b times
your second entry.

00:04:50.260 --> 00:04:53.000
And the whole reason why I'm
doing that is so that this guy

00:04:53.000 --> 00:04:55.820
zeroes out.

00:04:55.820 --> 00:04:59.730
So if we apply that to this
matrix up here-- let's do the

00:04:59.730 --> 00:05:01.610
first row first.

00:05:01.610 --> 00:05:05.680
So this first entry right here
is going to be ad minus bc

00:05:05.680 --> 00:05:08.725
times a, because that's C1--
let me write that down.

00:05:08.725 --> 00:05:20.330
So it's ad minus bc times a
minus b times C2 minus 0.

00:05:20.330 --> 00:05:22.210
So it's just going to be--
that second term

00:05:22.210 --> 00:05:24.050
just becomes 0.

00:05:24.050 --> 00:05:25.020
Fair enough.

00:05:25.020 --> 00:05:26.310
Now what is this guy
going to be?

00:05:26.310 --> 00:05:28.800
He's going to be-- I'll
write it out.

00:05:28.800 --> 00:05:40.440
He's going to be ad minus bc
times b minus b times your C2

00:05:40.440 --> 00:05:44.330
in this column vector, minus
b times ad minus bc.

00:05:47.040 --> 00:05:49.070
And you can see immediately that
these two guys are going

00:05:49.070 --> 00:05:52.370
to cancel out and you're
going to get a 0 there.

00:05:52.370 --> 00:05:54.340
And then we've got
to augment it.

00:05:54.340 --> 00:05:56.140
I want to make sure I don't run
out of space, I should've

00:05:56.140 --> 00:05:58.050
started to the left
a little bit more.

00:05:58.050 --> 00:05:59.640
So what's this guy
going to be?

00:05:59.640 --> 00:06:03.316
Well I'm going to have this guy
times ad minus bc-- I'll

00:06:03.316 --> 00:06:07.940
do it in pink-- so you're going
to have ad minus bc

00:06:07.940 --> 00:06:13.950
times 1, which is just ad minus
bc minus b times C2, so

00:06:13.950 --> 00:06:16.810
minus b times minus c.

00:06:16.810 --> 00:06:18.060
So that's plus bc.

00:06:20.450 --> 00:06:25.350
So 1 times ad minus bc
minus b times minus

00:06:25.350 --> 00:06:26.620
c is equal to that.

00:06:26.620 --> 00:06:28.460
And you can immediately see
that these two guys will

00:06:28.460 --> 00:06:28.990
cancel out.

00:06:28.990 --> 00:06:29.570
You're just [INAUDIBLE]

00:06:29.570 --> 00:06:30.590
ad.

00:06:30.590 --> 00:06:34.660
And then this guy over here,
you're going to have 0 times

00:06:34.660 --> 00:06:39.440
ad minus bc, which is just
a 0, minus b times a.

00:06:39.440 --> 00:06:44.070
So you have minus ab-- just
squeeze it in there.

00:06:44.070 --> 00:06:47.240
And we know that our second
row just stays the same.

00:06:47.240 --> 00:06:50.680
Our second row just stays the
same in this transformation.

00:06:50.680 --> 00:06:52.580
So we had a 0 here, we're
still going to have a 0.

00:06:52.580 --> 00:06:54.740
We had an ad minus bc.

00:06:54.740 --> 00:06:56.880
We'll still have
an ad minus bc.

00:06:56.880 --> 00:06:58.950
We had a minus c.

00:06:58.950 --> 00:07:01.300
Then we had an a.

00:07:01.300 --> 00:07:02.470
Just like that.

00:07:02.470 --> 00:07:05.640
Now let me re-write this matrix
just so it gets cleaned

00:07:05.640 --> 00:07:06.430
up a little bit.

00:07:06.430 --> 00:07:09.100
So let me re-write
it right here.

00:07:09.100 --> 00:07:11.150
I'll do it in my orange--
well, let me

00:07:11.150 --> 00:07:14.910
do it in this yellow.

00:07:14.910 --> 00:07:21.310
So I have ad minus bc times a.

00:07:21.310 --> 00:07:24.490
And then this term right
here just became a 0.

00:07:24.490 --> 00:07:26.200
This term right here is a 0.

00:07:26.200 --> 00:07:29.010
This term right here
is an ad minus bc.

00:07:31.890 --> 00:07:35.800
And then our augmented part,
this part was just an ad.

00:07:35.800 --> 00:07:38.860
This was a minus ab.

00:07:38.860 --> 00:07:40.660
This is a minus c.

00:07:40.660 --> 00:07:43.310
And then this is an a.

00:07:43.310 --> 00:07:46.460
Now we're almost at reduced row
echelon form right here.

00:07:46.460 --> 00:07:49.550
These two things just have to be
equal to 1 in order to get

00:07:49.550 --> 00:07:50.850
reduced row echelon form.

00:07:50.850 --> 00:07:53.190
So let's define a transformation
that'll make

00:07:53.190 --> 00:07:54.910
both of these equal to 1.

00:07:54.910 --> 00:07:58.680
So if this was T2 let me define
my transformation T3.

00:08:02.570 --> 00:08:06.170
You give it a column
vector, C1, C2.

00:08:06.170 --> 00:08:08.740
And it's just going to scale
each of the column vectors.

00:08:08.740 --> 00:08:13.650
So what I want to do is I want
to divide my first entries by

00:08:13.650 --> 00:08:16.990
this scaling factor right here
so that this becomes a 1.

00:08:16.990 --> 00:08:25.450
So I'm essentially going to
multiply 1 over ad minus bc

00:08:25.450 --> 00:08:30.870
times a, so 1 over ad, bc, a
times my first entry in each

00:08:30.870 --> 00:08:33.020
of my column vectors.

00:08:33.020 --> 00:08:34.809
And then my second one I
want to divide by this.

00:08:34.809 --> 00:08:36.600
So that this guy becomes a 1.

00:08:36.600 --> 00:08:39.120
So I'm doing two scalar
divisions in one

00:08:39.120 --> 00:08:40.020
transformation.

00:08:40.020 --> 00:08:47.700
So this one's going to be 1
over ad minus bc times C2.

00:08:47.700 --> 00:08:50.810
So I'm just scaling everything
by these two scaling factors.

00:08:50.810 --> 00:08:52.840
So if you apply this
transformation to that right

00:08:52.840 --> 00:08:54.820
there, what do we get?

00:08:54.820 --> 00:08:56.960
We get a matrix.

00:08:56.960 --> 00:09:01.920
And this guy, I'm going to
divide him by ad minus bc

00:09:01.920 --> 00:09:04.010
times a, so I'm dividing
it by itself, so that

00:09:04.010 --> 00:09:05.810
guy's going to be 1.

00:09:05.810 --> 00:09:08.340
I'm going to divide 0 by
this, but 0 divided by

00:09:08.340 --> 00:09:12.360
anything is just 0.

00:09:12.360 --> 00:09:13.940
Then we're in an
augmented part.

00:09:13.940 --> 00:09:17.340
ad divided by-- so let me
write it like this.

00:09:17.340 --> 00:09:22.120
So ad-- I'm going to divide by
this-- so it's going to be ad

00:09:22.120 --> 00:09:26.060
minus bc times a-- you
immediately see that the a's

00:09:26.060 --> 00:09:27.450
cancel out.

00:09:27.450 --> 00:09:34.180
This is going to be minus
ab divided by ad

00:09:34.180 --> 00:09:36.640
minus bc times a.

00:09:36.640 --> 00:09:38.900
Once again, the a's
cancel out.

00:09:38.900 --> 00:09:42.150
And then in my second row are my
second entries in my column

00:09:42.150 --> 00:09:45.940
vectors, 0 divided
by anything is 0.

00:09:45.940 --> 00:09:49.290
So 0 divided by this thing is
going to be 0, assuming we can

00:09:49.290 --> 00:09:51.020
divide by that, and we're
going to talk

00:09:51.020 --> 00:09:53.050
about that in a second.

00:09:53.050 --> 00:09:55.840
This guy divided by this guy,
we're just dividing by

00:09:55.840 --> 00:09:58.180
himself, so it's going
to be equal to 1.

00:09:58.180 --> 00:10:06.190
Now we have minus c divided
by this, or ad minus bc.

00:10:06.190 --> 00:10:08.470
And then we have an a.

00:10:08.470 --> 00:10:13.370
a divided by ad, ad minus bc.

00:10:13.370 --> 00:10:14.280
And we're done.

00:10:14.280 --> 00:10:18.740
We put the left-hand side of
our augmented matrix into

00:10:18.740 --> 00:10:20.410
reduced row echelon form.

00:10:20.410 --> 00:10:22.080
And now this is going
to be our inverse.

00:10:22.080 --> 00:10:25.260
So let me clean it
up a little bit.

00:10:25.260 --> 00:10:28.270
So, so far we started off with
a matrix-- I'll do it in

00:10:28.270 --> 00:10:31.830
purple-- we started off
with a matrix a is

00:10:31.830 --> 00:10:35.830
equal to a, b, c, d.

00:10:35.830 --> 00:10:38.510
And now just using our technique
we figured out that

00:10:38.510 --> 00:10:42.740
a inverse is equal to this
thing right here.

00:10:42.740 --> 00:10:45.020
And just to simplify-- well let
me just write it the way I

00:10:45.020 --> 00:10:46.705
have it there, because I don't
want to skip any steps-- this

00:10:46.705 --> 00:10:52.920
is equal to d over
ad minus bc.

00:10:52.920 --> 00:10:55.670
Right, this guy and that
guy canceled out.

00:10:55.670 --> 00:11:02.290
And then we have a minus b over
ad minus bc because that

00:11:02.290 --> 00:11:04.110
guy and that guy canceled out.

00:11:04.110 --> 00:11:09.080
Then you have a minus
c over ad minus bc.

00:11:09.080 --> 00:11:14.100
And then finally you have
an a over ad minus bc,

00:11:14.100 --> 00:11:15.430
which is our inverse.

00:11:15.430 --> 00:11:18.460
But one thing might just pop out
at you immediately is that

00:11:18.460 --> 00:11:21.760
everything in our inverse is
being divided by this.

00:11:21.760 --> 00:11:24.640
So maybe an easier way
to write our inverse.

00:11:24.640 --> 00:11:26.290
We could also write our
inverse like this.

00:11:26.290 --> 00:11:31.950
We could just write it as 1
over ad minus bc times the

00:11:31.950 --> 00:11:38.800
matrix d minus b
minus c and a.

00:11:38.800 --> 00:11:42.550
And just like that we have come
up with a formula for the

00:11:42.550 --> 00:11:44.290
inverse of a 2 by 2 matrix.

00:11:44.290 --> 00:11:48.320
You give me any real numbers
here and I'm going to give you

00:11:48.320 --> 00:11:48.970
its inverse.

00:11:48.970 --> 00:11:50.280
That straightforward.

00:11:50.280 --> 00:11:53.510
Now one thing you might be
saying, hey, but not all 2 by

00:11:53.510 --> 00:11:55.620
2 matrices are invertible.

00:11:55.620 --> 00:11:58.390
How can this be the case
for all of them.

00:11:58.390 --> 00:12:00.740
And I'll give you a question,
when will this thing right

00:12:00.740 --> 00:12:02.940
here not be defined?

00:12:02.940 --> 00:12:06.520
When is this thing
not defined?

00:12:06.520 --> 00:12:09.710
Every operation I did, I can do
with any real numbers, and

00:12:09.710 --> 00:12:10.930
this applies to any
real numbers.

00:12:10.930 --> 00:12:12.240
But when is this thing
not defined?

00:12:16.230 --> 00:12:19.160
Well it's not defined
when I divide by 0.

00:12:19.160 --> 00:12:20.900
And when would I divide by 0?

00:12:20.900 --> 00:12:23.650
Everything else you can multiply
and subtract and add

00:12:23.650 --> 00:12:26.330
a zero to anything, but you
just can't divide by 0.

00:12:26.330 --> 00:12:28.980
We've never defined what it
means when you divide

00:12:28.980 --> 00:12:30.000
something by 0.

00:12:30.000 --> 00:12:38.060
So it's not defined if ad
minus bc is equal to 0.

00:12:38.060 --> 00:12:39.530
So this is an interesting
thing.

00:12:39.530 --> 00:12:44.470
I can always find the inverse of
a 2 by 2 matrix as long as

00:12:44.470 --> 00:12:48.590
ad minus bc is not equal to 0.

00:12:48.590 --> 00:12:50.510
We came up with all of these
fancy things for

00:12:50.510 --> 00:12:52.120
invertability, you've got to
put it into reduced row

00:12:52.120 --> 00:12:54.410
echelon form, and before
that we talked being

00:12:54.410 --> 00:12:56.150
on 2 and 1 to 1.

00:12:56.150 --> 00:12:57.870
For at least a 2 by
2 matrix we've

00:12:57.870 --> 00:12:59.440
really simplified things.

00:12:59.440 --> 00:13:07.220
As long as ad minus bc does not
equal 0, we can use this

00:13:07.220 --> 00:13:11.960
formula and then we know that
a-- and it goes both ways-- a

00:13:11.960 --> 00:13:13.470
is invertible.

00:13:13.470 --> 00:13:16.930
And not only is it invertible,
but we can just apply this

00:13:16.930 --> 00:13:18.200
formula to it.

00:13:18.200 --> 00:13:20.900
So immediately something
interesting might-- you might

00:13:20.900 --> 00:13:24.000
say hey, this is an interesting
number.

00:13:24.000 --> 00:13:25.990
We should come up with
some name for it.

00:13:25.990 --> 00:13:29.620
And lucky for us, we have come
up with a name for it.

00:13:29.620 --> 00:13:30.870
This is called the
determinant.

00:13:33.410 --> 00:13:35.650
Let me write it in pink.

00:13:35.650 --> 00:13:36.900
Determinant.

00:13:39.220 --> 00:13:45.670
So the determinant of a, and
it's also written like this

00:13:45.670 --> 00:13:49.580
with these little straight lines
around a, and you could

00:13:49.580 --> 00:13:54.700
also write it like
this, a, b, c, d.

00:13:54.700 --> 00:13:57.040
But most people kind of think
this is redundant to have

00:13:57.040 --> 00:13:58.830
brackets and these lines.

00:13:58.830 --> 00:14:01.070
So then they just write it like
this, this is equal to

00:14:01.070 --> 00:14:04.650
just, they just write the
lines, a, b, c, d.

00:14:04.650 --> 00:14:05.670
I want to make this
very clear.

00:14:05.670 --> 00:14:08.250
If you have the brackets you're
dealing with a matrix.

00:14:08.250 --> 00:14:10.410
If you have just these straight
lines you're talking

00:14:10.410 --> 00:14:12.740
about the determinant
of the matrix.

00:14:12.740 --> 00:14:16.930
But this is defined for
the 2 by 2 case to be

00:14:16.930 --> 00:14:20.920
equal to ad minus bc.

00:14:20.920 --> 00:14:28.050
This is a definition
of the determinant.

00:14:28.050 --> 00:14:32.100
So we can re-write, if we have
some matrix here, we have some

00:14:32.100 --> 00:14:37.730
matrix a which is equal
to a, b, c, d.

00:14:37.730 --> 00:14:43.690
We can now write its inverse,
a inverse is equal to 1 over

00:14:43.690 --> 00:14:45.920
this thing, which we've defined
as the determinant of

00:14:45.920 --> 00:14:54.570
a times-- and let's just
see a good way of kind

00:14:54.570 --> 00:14:55.500
of memorizing this.

00:14:55.500 --> 00:14:59.940
We're swapping these two
guys, right, the a

00:14:59.940 --> 00:15:01.000
and the d get swapped.

00:15:01.000 --> 00:15:03.390
So you get a d and an a.

00:15:03.390 --> 00:15:05.050
And then these two guys
stay the same,

00:15:05.050 --> 00:15:06.130
they just become negative.

00:15:06.130 --> 00:15:09.800
So minus b and minus c.

00:15:09.800 --> 00:15:11.890
So that's the general
formula for the

00:15:11.890 --> 00:15:13.600
determinant of a 2 by 2 matrix.

00:15:13.600 --> 00:15:15.350
Let's try to do a couple.

00:15:15.350 --> 00:15:20.500
Let's try to find the
determinant of the

00:15:20.500 --> 00:15:26.270
matrix 1, 2, 3, 4.

00:15:26.270 --> 00:15:27.190
Easy enough.

00:15:27.190 --> 00:15:30.750
So the determinant of-- let's
say this is the matrix B.

00:15:30.750 --> 00:15:33.040
So the determinant of B, or we
could write it like that,

00:15:33.040 --> 00:15:36.270
that's equal to the
determinant of B.

00:15:36.270 --> 00:15:39.320
That is just equal to-- that's
this thing right here-- 1

00:15:39.320 --> 00:15:49.300
times 4 minus 3 times 2, which
is equal to 4 minus 6, which

00:15:49.300 --> 00:15:50.480
is equal to minus 2.

00:15:50.480 --> 00:15:52.760
So the determinant is minus
2, so this is invertible.

00:15:55.430 --> 00:15:57.810
Not only is it invertible, but
it's very easy to find its

00:15:57.810 --> 00:15:58.330
inverse now.

00:15:58.330 --> 00:16:00.990
We can apply this formula.

00:16:00.990 --> 00:16:04.020
The inverse of B in this case--
let me do it in this

00:16:04.020 --> 00:16:07.635
color-- B inverse is equal to
1 over the determinant, so

00:16:07.635 --> 00:16:12.650
it's 1 over minus 2 times the
matrix where we swap-- well,

00:16:12.650 --> 00:16:14.725
this is the determinant of B.

00:16:14.725 --> 00:16:15.720
I want to be careful.

00:16:15.720 --> 00:16:18.980
B is the same thing,
but with brackets.

00:16:18.980 --> 00:16:22.190
1, 2, 3, 4.

00:16:22.190 --> 00:16:25.410
So B inverse is going to be 1
over the determinant of B,

00:16:25.410 --> 00:16:27.510
which is equal to minus 2.

00:16:27.510 --> 00:16:28.880
So 1 over minus 2.

00:16:28.880 --> 00:16:32.430
We swap these two guys, so they
get a 4 and a 1, and then

00:16:32.430 --> 00:16:35.150
these two guys become
negative-- minus 2

00:16:35.150 --> 00:16:36.970
and then minus 3.

00:16:36.970 --> 00:16:39.010
And then if we were to multiply
this out it would be

00:16:39.010 --> 00:16:43.920
equal to minus 1/2 times
4 is minus 2.

00:16:43.920 --> 00:16:46.980
Minus 1/2 times minus 2 is 1.

00:16:46.980 --> 00:16:51.220
Minus 1/2 times minus
3 is 3/2, minus 1/2

00:16:51.220 --> 00:16:53.610
times 1 is minus 1/2.

00:16:53.610 --> 00:16:56.010
So that there is the
inverse of B.

00:16:56.010 --> 00:16:57.780
Now let's say we have
another matrix.

00:16:57.780 --> 00:17:00.060
Let's say we have the
matrix C, and C is

00:17:00.060 --> 00:17:05.400
equal to 1, 2, 3, 6.

00:17:05.400 --> 00:17:09.390
What is the determinant of C?

00:17:09.390 --> 00:17:12.710
It is equal to-- we could write
this way-- 1, 2, 3, 6.

00:17:15.540 --> 00:17:21.280
And it is equal to 1 times 6
minus 3 times 2, which is

00:17:21.280 --> 00:17:25.079
equal to 6 minus 6, which
is equal to 0.

00:17:25.079 --> 00:17:29.110
And there you see it's equal to
0, so you cannot find-- so

00:17:29.110 --> 00:17:30.360
this is not invertible.

00:17:35.090 --> 00:17:37.170
So we can't find its inverse
because if we would try to

00:17:37.170 --> 00:17:40.550
apply this formula right here
you'd have a 1 over 0.

00:17:40.550 --> 00:17:43.180
But we know this formula just
comes out-- that attempt to

00:17:43.180 --> 00:17:45.570
put it into reduced row echelon
form, and in that last

00:17:45.570 --> 00:17:48.600
step we just had to essentially
divide everything

00:17:48.600 --> 00:17:52.540
by these terms. So these terms
would be 0 in this matrix C

00:17:52.540 --> 00:17:55.430
that I just constructed
for you.

00:17:55.430 --> 00:17:58.510
And the reason why I knew-- I
just pulled this out of my

00:17:58.510 --> 00:18:00.810
brain-- I knew this wasn't going
to be invertible because

00:18:00.810 --> 00:18:03.410
I constructed a situation where
I have columns that are

00:18:03.410 --> 00:18:04.820
linear combinations
of each other.

00:18:04.820 --> 00:18:08.950
I have 1, 3-- you multiply that
by 2 you get 2 and 6.

00:18:08.950 --> 00:18:11.900
So I knew that these aren't
linearly independent columns.

00:18:11.900 --> 00:18:14.250
So you know that its rank wasn't
going to be equal to,

00:18:14.250 --> 00:18:16.800
so I knew it wasn't going to be
invertible, but we see that

00:18:16.800 --> 00:18:19.640
here by just computing
its determinant.