Duplicate row determinant

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:05.270
Say I have some matrix a --
let's say a is n by n, so it

00:00:05.270 --> 00:00:07.390
looks something like this.

00:00:07.390 --> 00:00:10.990
You've seen this before,
a 1 1, a 1 2, all

00:00:10.990 --> 00:00:13.690
the way to a 1 n.

00:00:13.690 --> 00:00:16.980
When you go down the rows you
get a 2 1, that goes all the

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way to a 2 n.

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And let's say that there's some
row here, let's say row

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i, it looks like a i 1,
all the way to a i n.

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And then you have some other row
here, a j, it's a j 1 all

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the way to a j n.

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And then you keep going all the
way down to a n 1, a n 2,

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all the way to a n n.

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This is just an n by n matrix,
and you can see that I took a

00:00:48.070 --> 00:00:54.030
little trouble to write out my
row a, my i'th row here and my

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j'th row here.

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And just to kind of keep things
a little simple, let me

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just define -- just for
notational purposes, you can

00:01:02.850 --> 00:01:05.480
view these as row vectors if
you like, but I haven't

00:01:05.480 --> 00:01:07.270
formally defined row
vectors so I won't

00:01:07.270 --> 00:01:08.890
necessarily go there.

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But let's just define the term r
i, we'll call that row i, to

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be equal to a i 1, a i 2,
all the way to a i n.

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You can write it as
a vector if you

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like, like a row vector.

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We haven't really defined
operations on row vectors that

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well yet, but I think
you get the idea.

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We can then replace this guy
with r 1, this guy with r 2,

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all the way down.

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Let me do that, and I'll do
that in the next couple of

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videos because it'll simplify
things, and I think make

00:01:39.830 --> 00:01:41.710
things a little bit easier
to understand.

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So I can rewrite this matrix,
this n by n matrix a, I can

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re-write it as just r i.

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Actually, this just looks
like a vector,

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it's just a row vector.

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Let me write it as a
vector like that.

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And I'm being a little bit
hand-wavy here because all of

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our vectors have been defined as
column vectors, but I think

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you get the idea.

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So let's call that r 1, and then
we have r 2 is the next

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row, all the way down.

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You keep going down, you get
to r i -- that's this row

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right there -- r i.

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You keep going down, you get r
j, and then you keep going

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down until you get
to the n'th row.

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And each of these guys are going
to have n terms because

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you have n columns.

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So that's another
way of writing

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this same n by n matrix.

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Now what I'm going to do here
is, I'm going to create a new

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matrix-- let's call that
swapping the swap

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matrix of i and j.

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So I'm going to swap i and
j, those two rows.

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So what's the matrix
going to look like?

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Everything else is going
to be equal.

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You have row 1-- assuming that
1 wasn't one of the i or j's,

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it could have been.

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Row 2, all the way down to-- now
instead of a row i there

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you have a row j there, and you
go down and instead of a

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row j you have a row i there.

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And you go down and
then you get r n.

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So what did we do?

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We just swapped these
two guys.

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That's what the swap
matrix is.

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Now I think it was in the last
video or a couple of videos

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ago, we learned that if you just
swap two rows of any n by

00:03:22.530 --> 00:03:28.020
n matrix, the determinant of the
resulting matrix will be

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the negative of the original
determinant.

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So we get the determinant of
s, the swap of the i'th and

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the j rows is going to be equal
to the minus of the

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determinant of a.

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Now, let me ask you an
interesting question.

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What happens if those two rows
were actually the same?

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What if r i was equal to r j?

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If we go back to all of these
guys, if that row is

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equal to this row?

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That means that this guy is
equal to that guy, that the

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second column-- the second
column for that row all the

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way to the n'th guy is equal
to the n'th guy.

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That's what I mean when I say
what happens if those two rows

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are equal to each other.

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Well, if those two rows are
equal to each other, than this

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matrix is no different than this
matrix here, even though

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we swapped them.

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If you swap two identical
things, you're just going to

00:04:27.340 --> 00:04:29.640
be left with the same
thing again.

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So if-- let me write this down--
if row i is equal to

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row j, then this guy,
then s, the swapped

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matrix, is equal to a.

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They'll be identical.

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You're swapping two rows that
are the same thing.

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So that implies a determinant of
the swapped matrix is equal

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to the determinant of a.

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But we just said, if the swap
matrix, when you swap two

00:05:01.250 --> 00:05:03.800
rows, it equals a negative
of the determinant of a.

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So this tells us it also has to
equal the negative of the

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determinant of a.

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So what does that tell us?

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That tells us if a has two rows
that are equal to each

00:05:14.840 --> 00:05:20.320
other, if we swap them, we
should get the negative of the

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determinant, but if two rows are
equal we're going to get

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the same matrix again.

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So if a has two rows that are
equal-- so if row i is equal

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to row j-- then the determinant
of a has to be

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equal to the negative of
the determinant of a.

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We know that because the
determinant of a, or a is the

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same thing as the swapped
version of a, and the swapped

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version of a has to have the
negative determinant of a.

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So these two things
have to be equal.

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Now what number is equal to a
negative version of itself?

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If I just told you x is equal
to negative x, what number

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does x have to be equal to?

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There's only one value that it
could possibly be equal to. x

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would have to be equal to 0.

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So the takeaway here is, let's
say if you have duplicate

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rows-- you can extend this if
you have three or four rows

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that are the same-- leads
you to the fact that the

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determinant of your
matrix is 0.

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And that really shouldn't
be a surprise.

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Because if you have duplicate
rows, remember what we learned

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a long time ago.

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We learned that a matrix is an
invertible if and only if the

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reduced row echelon form
is the identity matrix.

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We learned that.

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But if you have two duplicate
rows-- let's say these two

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guys are equal to each other--
you could perform a row

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operation where you replace this
guy with this guy minus

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that guy, and you'll just
get a row of 0's.

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And if you get a row of 0's,
you're never going to be able

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get the identity matrix.

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So we know that duplicate rows
could never get reduced row

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echelon form to be
the identity.

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Or, duplicate rows are
not invertible.

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And we also learned that
something is not invertible if

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and only if its determinant
is equal to 0.

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So we now got to the same result
two different ways.

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One, we just used some
of what we learned.

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When you swap rows, it should
become the negative, but if

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you swap the same row, you
shouldn't change the matrix.

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So the determinant of
the matrix has to

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be the same as itself.

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So if you have duplicate rows,
the determinant is 0.

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Which isn't something that we
had to use using this little

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swapping technique, we could
have gone back to our

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requirements for invertability--
I think was

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five or six videos ago.

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But I just wanted to
point that out.

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If you see duplicate rows.

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and actually if you see
duplicate columns-- I'll leave

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that for you to think about--
if you see duplicate rows or

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duplicate columns, or even if
you just see that some rows

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are linear combinations of
other rows-- and I'm not

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showing that to you right here--
then you know that your

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determinant is going
to be equal to 0.


Determinant when row is added

WEBVTT
Kind: captions
Language: en

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Let's keep messing with our
determinants to see if we can

00:00:02.990 --> 00:00:04.910
get more useful results.

00:00:04.910 --> 00:00:07.140
And they might not be obviously
useful right now,

00:00:07.140 --> 00:00:10.350
but maybe we'll use them later
when we are exploring other

00:00:10.350 --> 00:00:12.060
parts of linear algebra.

00:00:12.060 --> 00:00:15.090
So let's say I have
some matrix, let's

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call it matrix X.

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Matrix X is equal to-- I'll just
start with a 3 by 3 case

00:00:21.710 --> 00:00:24.100
because I think the 2 by 2
case is a bit trivial.

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Actually, why don't I just
start with a 2 by 2 case.

00:00:26.210 --> 00:00:32.020
Let's say matrix X is a, b,
and then it has x1, x2.

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I could have called these c and
d, but you'll see why I

00:00:34.030 --> 00:00:36.240
called them x1 and
x2 in a second.

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And I'll say I have
another matrix.

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Let's say matrix Y is identical
to matrix X except

00:00:42.370 --> 00:00:43.610
for this row.

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So matrix Y is a, b y1 and y2.

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And let's say we have
a third matrix Z.

00:00:55.690 --> 00:01:00.070
That's identical to the first
two matrices on the first row.

00:01:00.070 --> 00:01:01.290
So a, b.

00:01:01.290 --> 00:01:04.790
but on the second row it's
actually the sum of the two

00:01:04.790 --> 00:01:06.150
rows of x and y.

00:01:06.150 --> 00:01:09.070
So it's going to be, this
entry's going to be x1 plus

00:01:09.070 --> 00:01:18.910
y1, and this entry right
here is x2 plus y2.

00:01:18.910 --> 00:01:20.150
Just like that.

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I want to be very clear,
Z is not X plus Y.

00:01:23.050 --> 00:01:25.800
All of the terms of Z are
not the sum of all the

00:01:25.800 --> 00:01:26.690
terms of X and Y.

00:01:26.690 --> 00:01:29.770
I'm only focusing on
one particular row.

00:01:29.770 --> 00:01:31.990
And this is just a general theme
that you'll see over and

00:01:31.990 --> 00:01:33.920
over again, and we saw it in
the last video and I guess

00:01:33.920 --> 00:01:37.790
you'll see it here, is that
determinants or finding the

00:01:37.790 --> 00:01:41.530
determinants of matrices
aren't linear on matrix

00:01:41.530 --> 00:01:44.720
operations, but they are linear
on operations that you

00:01:44.720 --> 00:01:46.330
do just to one row.

00:01:46.330 --> 00:01:49.680
So in this case, everything else
is equal except for this

00:01:49.680 --> 00:01:54.000
row, and Z has the same first
row as these guys, but its

00:01:54.000 --> 00:01:57.080
second row is the sum of the
second row of these guys.

00:01:57.080 --> 00:01:59.090
So let's explore how
the determinants

00:01:59.090 --> 00:02:01.130
of these guys relate.

00:02:01.130 --> 00:02:05.270
So the determinant-- let
me do it in X's color.

00:02:05.270 --> 00:02:08.639
The determinant of X-- I'll
write it like that-- is equal

00:02:08.639 --> 00:02:13.200
to a ax2 minus bx1.

00:02:13.200 --> 00:02:15.070
You've seen that
multiple times.

00:02:15.070 --> 00:02:22.670
The determinant of Y is equal
to ay2 minus by1.

00:02:22.670 --> 00:02:31.130
And the determinant of Z is
equal to a times x2 plus y2

00:02:31.130 --> 00:02:38.150
minus b times x1 plus y1, which
is equal to ax2 plus

00:02:38.150 --> 00:02:45.150
ay2-- just distributed the
a-- minus bx1 minus by1.

00:02:45.150 --> 00:02:48.710
And if we just rearrange things,
this is equal to a--

00:02:48.710 --> 00:02:50.510
let me write it this
way-- this is equal

00:02:50.510 --> 00:02:54.890
to ax2 minus bx1.

00:02:54.890 --> 00:02:58.740
That's that term and that term,
we switched colors.

00:02:58.740 --> 00:03:00.560
So that's those two guys.

00:03:00.560 --> 00:03:07.320
And then plus ay2 minus by1.

00:03:07.320 --> 00:03:09.190
Now what is this right here?

00:03:09.190 --> 00:03:10.900
That is the determinant of X.

00:03:13.680 --> 00:03:17.290
And this right here is
the determinant of Y.

00:03:17.290 --> 00:03:18.190
So there you have it.

00:03:18.190 --> 00:03:22.190
If we have matrices that are
completely identical except

00:03:22.190 --> 00:03:25.160
for one row-- and in this case
it's a 2 by 2 matrix, so it

00:03:25.160 --> 00:03:29.115
looks like half of the matrix--
and Z's, that row

00:03:29.115 --> 00:03:32.080
that we're referring to that's
different, Z's is the sum of

00:03:32.080 --> 00:03:37.530
the other two guys' rows, then
Z's determinant is the sum of

00:03:37.530 --> 00:03:38.590
the other two determinants.

00:03:38.590 --> 00:03:39.990
So this is a very
special case.

00:03:39.990 --> 00:03:40.840
I want to keep reiterating it.

00:03:40.840 --> 00:03:44.530
It only works in the case where
this row and only this

00:03:44.530 --> 00:03:48.740
row is the sum of this row and
this row, and the matrices are

00:03:48.740 --> 00:03:50.850
identical everywhere else.

00:03:50.850 --> 00:03:54.490
Let me show you the 3 by 3 case,
and I think it'll be a

00:03:54.490 --> 00:03:55.290
little bit more general.

00:03:55.290 --> 00:03:56.420
And then we'll go to n by n.

00:03:56.420 --> 00:03:58.500
The n by n is actually, on some
level, the easiest to do,

00:03:58.500 --> 00:04:02.430
but it's kind of abstract so I
like to save that for the end.

00:04:02.430 --> 00:04:08.300
So let's redefine all those
guys into the 3 by 3 case.

00:04:08.300 --> 00:04:17.029
So let's say that X is equal
to a, b, c-- let's just do,

00:04:17.029 --> 00:04:19.019
let's make the third row the
row we're going to use to

00:04:19.019 --> 00:04:23.810
determine our determinant. a,
b, c, d, e, f-- actually let

00:04:23.810 --> 00:04:25.230
me do the middle row, because I
don't want to make you think

00:04:25.230 --> 00:04:26.670
it always has to be
the last row.

00:04:26.670 --> 00:04:33.260
So let's say it's x1, c2, x3,
and you have d, e, f.

00:04:33.260 --> 00:04:34.910
And what's the determinant
of X going to be?

00:04:34.910 --> 00:04:39.200
The determinant of X is going
to be equal to-- let's say

00:04:39.200 --> 00:04:42.380
we're going along this
row right here,

00:04:42.380 --> 00:04:43.530
that's the row in question.

00:04:43.530 --> 00:04:45.690
It's going to be equal to--
well you remember your

00:04:45.690 --> 00:04:47.870
checkerboard pattern-- so it's
going to be-- remember, plus,

00:04:47.870 --> 00:04:50.080
minus, plus, minus, plus--
you remember all

00:04:50.080 --> 00:04:51.250
the rest how it goes.

00:04:51.250 --> 00:04:57.910
So it's going to start with
a minus x1 times the sub

00:04:57.910 --> 00:05:00.810
matrix-- you get rid of that
column, that row-- b, c, e, f.

00:05:04.470 --> 00:05:09.520
Then you have plus x2 times the
sub matrix-- get rid of

00:05:09.520 --> 00:05:12.140
that column, that row--
a, c, d, f.

00:05:15.580 --> 00:05:20.200
And then finally minus x3--
you get rid of its row and

00:05:20.200 --> 00:05:21.690
column-- you have a, b, d, e.

00:05:26.170 --> 00:05:31.190
Now let me define another matrix
Y that is identical to

00:05:31.190 --> 00:05:34.260
matrix X, except for that row.

00:05:34.260 --> 00:05:37.340
So it's a, b, c.

00:05:37.340 --> 00:05:39.210
Then down here d, e, f.

00:05:39.210 --> 00:05:40.250
That middle row is different.

00:05:40.250 --> 00:05:44.410
It's y1, y2, and y3.

00:05:44.410 --> 00:05:47.490
What's the determinant
of Y going to be?

00:05:47.490 --> 00:05:49.920
Determinant of Y?

00:05:49.920 --> 00:05:51.390
Well it's going to be identical
to the determinant

00:05:51.390 --> 00:05:53.510
of X because all the
sub-matrices are going to be

00:05:53.510 --> 00:05:54.870
the same when you cross
out this row

00:05:54.870 --> 00:05:56.120
and each of the columns.

00:05:56.120 --> 00:05:57.660
But the coefficients are
going to be different.

00:05:57.660 --> 00:05:59.390
Instead of an x1,
you have a y1.

00:05:59.390 --> 00:06:04.460
So it's going to be equal to
minus y1 times the determinant

00:06:04.460 --> 00:06:13.320
b, c, e, f plus y 2 times the
determinant of a, c, d f minus

00:06:13.320 --> 00:06:18.210
y3 times the determinant
of a, b, d, e.

00:06:18.210 --> 00:06:19.990
I think you see where
this is going.

00:06:19.990 --> 00:06:22.330
Now I'm going to create
another matrix.

00:06:22.330 --> 00:06:28.680
I'm going to create another
matrix Z just like that that

00:06:28.680 --> 00:06:33.290
is equal to-- it's identical to
these two guys on the first

00:06:33.290 --> 00:06:38.590
and third rows, a,
b, c, d, e, f.

00:06:38.590 --> 00:06:39.840
Just like that.

00:06:39.840 --> 00:06:43.060
But this row just happens
to be the sum of this

00:06:43.060 --> 00:06:44.233
row and this row.

00:06:44.233 --> 00:06:46.420
And when we figured out this
determinant we went along that

00:06:46.420 --> 00:06:48.390
row-- you can see that
right there.

00:06:48.390 --> 00:06:52.240
So this row right here is going
to be x1 plus y1, that's

00:06:52.240 --> 00:06:53.750
its first term.

00:06:53.750 --> 00:06:58.310
x2 plus y2, and then you
have x3 plus y3.

00:06:58.310 --> 00:07:01.110
Now what's the determinant
of Z going to be?

00:07:01.110 --> 00:07:04.010
Well, we can go down this
row right there.

00:07:04.010 --> 00:07:10.020
So it's going to be minus x1
plus y1 times its sub-matrix--

00:07:10.020 --> 00:07:12.730
get rid of that row, that
column-- you get b, c, e, f.

00:07:16.420 --> 00:07:18.470
I think you definitely see
where this is going.

00:07:18.470 --> 00:07:23.950
Plus this coefficient, plus x2
plus y2 times its sub-matrix--

00:07:23.950 --> 00:07:26.310
get rid of that row, that
column-- a, c, d, f.

00:07:30.610 --> 00:07:37.220
And then you have minus this
guy right here, x3 plus y3

00:07:37.220 --> 00:07:39.820
times its sub matrix-- get
rid of that column and

00:07:39.820 --> 00:07:41.130
row-- a, b, d, e.

00:07:45.370 --> 00:07:46.580
Now what do you have
right here?

00:07:46.580 --> 00:07:48.043
This is the determinant of Z.

00:07:50.700 --> 00:07:54.790
This right here is the
determinant of Z.

00:07:54.790 --> 00:07:55.970
This thing right here.

00:07:55.970 --> 00:07:57.430
I think you can see immediately
that if you were

00:07:57.430 --> 00:08:00.650
to add this to this you would
get this right here, right?

00:08:00.650 --> 00:08:02.900
Because you have this
coefficient and this

00:08:02.900 --> 00:08:03.670
coefficient on that.

00:08:03.670 --> 00:08:07.240
If you added them up you would
get minus x1 plus y1.

00:08:07.240 --> 00:08:11.080
This guy and this guy
add up to this guy.

00:08:11.080 --> 00:08:13.500
And then if I were to do
this guy and this guy,

00:08:13.500 --> 00:08:15.020
add up to that guy.

00:08:15.020 --> 00:08:16.830
Let me do another one.

00:08:16.830 --> 00:08:18.720
And then finally that
term plus that term

00:08:18.720 --> 00:08:20.040
add up to that term.

00:08:20.040 --> 00:08:22.790
So you immediately see that the
determinant, or hopefully

00:08:22.790 --> 00:08:29.580
you immediately see, that the
determinant of X plus the

00:08:29.580 --> 00:08:36.095
determinant of Y is equal
to the determinant of Z.

00:08:41.210 --> 00:08:43.350
So we did it for the 2 by
2 case, we just did it

00:08:43.350 --> 00:08:45.470
for the 3 by 3 case.

00:08:45.470 --> 00:08:47.670
Might as well do it for
the n by n case so we

00:08:47.670 --> 00:08:48.860
know that it works.

00:08:48.860 --> 00:08:51.380
But the argument is identical
to this 3 by 3 case.

00:08:51.380 --> 00:08:53.760
So that's good to keep in your
mind because 3 by 3 is easy to

00:08:53.760 --> 00:08:57.470
visualize, n by n is sometimes
a little bit abstract.

00:08:57.470 --> 00:09:01.240
So let me re-define
my matrices again.

00:09:01.240 --> 00:09:04.020
I'm just going to do the
same thing over again.

00:09:04.020 --> 00:09:05.660
So I'm going to have
a matrix X.

00:09:05.660 --> 00:09:09.370
But it's an n by n matrix.

00:09:12.140 --> 00:09:14.330
So let me write it this way.

00:09:14.330 --> 00:09:20.130
Let's say it is a 1, 1, a 1,
2, all the way to a 1, n.

00:09:20.130 --> 00:09:24.650
And there's some row here, let's
say that there's some

00:09:24.650 --> 00:09:29.010
row here on row i-- let's call
this row i right here-- and

00:09:29.010 --> 00:09:35.220
here it has the terms x1, x2,
all the way to xn, but

00:09:35.220 --> 00:09:37.490
everything else is just
the regular a's.

00:09:37.490 --> 00:09:40.830
So then you have a-- let
me make this as a21,

00:09:40.830 --> 00:09:42.310
all the way to a2n.

00:09:42.310 --> 00:09:46.710
And then if you went all the way
down here you would have

00:09:46.710 --> 00:09:51.150
an1, and you'd go all
the way to ann.

00:09:51.150 --> 00:09:53.380
So essentially you could imagine
our standard matrix

00:09:53.380 --> 00:09:56.360
where everything is defined in
a, but I replaced row i with

00:09:56.360 --> 00:09:59.400
certain numbers that are maybe
a little different.

00:09:59.400 --> 00:10:01.160
And I think you'll see
where I'm going.

00:10:01.160 --> 00:10:03.150
Now let me define
my other matrix.

00:10:03.150 --> 00:10:06.810
Let me define matrix Y.

00:10:06.810 --> 00:10:10.530
Let me define matrix Y to be
essentially the same thing.

00:10:10.530 --> 00:10:13.520
This is a11-- it's
the same a11.

00:10:13.520 --> 00:10:16.640
This is a12, all
the way to a1n.

00:10:16.640 --> 00:10:20.330
This is a21, we could go
all the way to a a2n.

00:10:20.330 --> 00:10:24.480
And then on row i, the same row,
this is n by n, this is

00:10:24.480 --> 00:10:27.950
the same n by n-- if this was
10 by 10, this is 10 by 10.

00:10:27.950 --> 00:10:30.680
If this is row seven, then
this is row seven.

00:10:30.680 --> 00:10:33.210
It has different terms. It's
identical to matrix X

00:10:33.210 --> 00:10:34.445
except for row i.

00:10:34.445 --> 00:10:39.890
In row i it is y1, y2,
all the way to yn.

00:10:39.890 --> 00:10:43.170
And if you keep going down, of
course, you have an1, all the

00:10:43.170 --> 00:10:46.350
way to ann.

00:10:46.350 --> 00:10:47.460
Fair enough.

00:10:47.460 --> 00:10:49.860
Now let's say we have
a third matrix.

00:10:49.860 --> 00:10:51.140
Let's have a third matrix.

00:10:51.140 --> 00:10:52.840
Let me draw it right here.

00:10:52.840 --> 00:10:57.190
So you have Z, Z is equal to--
I think you could imagine

00:10:57.190 --> 00:10:59.440
where this is going.

00:10:59.440 --> 00:11:04.460
Z is identical to these two
guys except for row i.

00:11:04.460 --> 00:11:06.160
So let me write that out.

00:11:06.160 --> 00:11:07.430
So Z looks like this.

00:11:07.430 --> 00:11:14.260
You have a11, a12, all
the way to a1n.

00:11:14.260 --> 00:11:18.670
And then you go down and then
row i happens to be the sum of

00:11:18.670 --> 00:11:21.770
the row i of matrix
X and matrix Y.

00:11:21.770 --> 00:11:31.510
So it is x1 plus y1, x2 plus y2,
all the way to xn plus yn.

00:11:31.510 --> 00:11:33.820
And then it you keep going
down, everything else is

00:11:33.820 --> 00:11:39.010
identical, an1 all
the way to ann.

00:11:39.010 --> 00:11:43.450
So all of these matrices are
identical except for row X has

00:11:43.450 --> 00:11:47.100
a different row i than
matrix Y does.

00:11:47.100 --> 00:11:50.810
And row Z is identical
everywhere except its row i is

00:11:50.810 --> 00:11:52.995
the sum of this row
i and that row i.

00:11:52.995 --> 00:11:55.420
So it's a very particular case,
but we can figure out

00:11:55.420 --> 00:11:56.870
their determinants.

00:11:56.870 --> 00:11:57.640
So what are the determinants?

00:11:57.640 --> 00:12:03.360
The determinant of X, the
determinant of matrix X-- and

00:12:03.360 --> 00:12:05.860
hopefully you're maybe a bit
comfortable with writing sigma

00:12:05.860 --> 00:12:08.110
notation, we did this
in the last matrix.

00:12:08.110 --> 00:12:15.460
We can go down this row right
there, and for each of these

00:12:15.460 --> 00:12:19.410
guys we can say so the
determinant is going to be

00:12:19.410 --> 00:12:21.135
equal to the sum.

00:12:24.420 --> 00:12:28.930
Let's say we start from j is
equal to 1-- j's going to be

00:12:28.930 --> 00:12:32.100
the column, so we're going to
take the sum of each of these

00:12:32.100 --> 00:12:36.600
terms from j is equal
to 1 to n.

00:12:36.600 --> 00:12:38.620
And then remember our
checkerboard pattern, so we

00:12:38.620 --> 00:12:40.130
don't know if this is a
positive or negative.

00:12:40.130 --> 00:12:45.650
We can figure it out by taking
negative 1 to the i plus j--

00:12:45.650 --> 00:12:49.350
remember, this is the ith row
that we're talking about--

00:12:49.350 --> 00:12:59.550
times xj-- xj is the
coefficient, xsubj, times the

00:12:59.550 --> 00:13:03.790
sub matrix for xsubj.

00:13:03.790 --> 00:13:05.950
So if you get rid of this
guy's row and this guy's

00:13:05.950 --> 00:13:07.570
column, what is it
going to be?

00:13:07.570 --> 00:13:11.020
We could say that that's the
same thing as the sub matrix,

00:13:11.020 --> 00:13:14.000
if we called this guy-- let me
write it this way-- if we got

00:13:14.000 --> 00:13:17.180
rid of this guy's row and this
guy's column, if we had just

00:13:17.180 --> 00:13:19.370
our traditional matrix where
this wasn't replaced.

00:13:19.370 --> 00:13:25.570
If we just had an ai1 here, ai2,
its sub matrix would be

00:13:25.570 --> 00:13:27.580
the same thing, because we're
crossing out this

00:13:27.580 --> 00:13:29.330
row and this column.

00:13:29.330 --> 00:13:31.360
So it would be all of these
guys and all of

00:13:31.360 --> 00:13:32.360
these guys down here.

00:13:32.360 --> 00:13:36.180
So it would be the sub matrix--
this is a n minus 1

00:13:36.180 --> 00:13:42.530
by n minus 1 matrix-- it would
be the sub matrix for aij.

00:13:42.530 --> 00:13:46.060
That's for the first term--
sorry, the determinant.

00:13:46.060 --> 00:13:48.890
Don't want to lose the
determinant there-- times the

00:13:48.890 --> 00:13:52.660
determinant of the
sub matrix aij.

00:13:52.660 --> 00:13:54.170
And so that's for the first
term, and then you're going to

00:13:54.170 --> 00:13:56.520
add it to the second term, and
then you're just going to keep

00:13:56.520 --> 00:13:57.030
doing that.

00:13:57.030 --> 00:13:58.650
That's what this sigma
notation is.

00:13:58.650 --> 00:14:00.890
That's the determinant of X.

00:14:00.890 --> 00:14:03.930
Now what's the determinant
of Y?

00:14:03.930 --> 00:14:09.940
The determinant of Y is equal
to the sum-- we could do the

00:14:09.940 --> 00:14:13.610
same thing-- j is equal to
1 to n of negative 1

00:14:13.610 --> 00:14:15.770
to the i plus j.

00:14:15.770 --> 00:14:19.710
We're going to go along this row
right here, the ith row.

00:14:19.710 --> 00:14:22.990
So we're going to have ysubj--
right, we're going to start

00:14:22.990 --> 00:14:27.840
with ysub1, then plus ysub2,
times the determinant of its

00:14:27.840 --> 00:14:29.640
sub matrix, which is
the same as the

00:14:29.640 --> 00:14:30.600
determinant of this sub matrix.

00:14:30.600 --> 00:14:32.820
So you get rid of that row and
that column for each of these

00:14:32.820 --> 00:14:35.480
guys, that everything else on
the matrix is the same.

00:14:35.480 --> 00:14:37.870
So aij.

00:14:37.870 --> 00:14:39.840
The matrix of aij.

00:14:39.840 --> 00:14:42.040
Now what is the determinant
of Z.

00:14:42.040 --> 00:14:44.930
I'm pretty sure you know exactly
where this is going.

00:14:44.930 --> 00:14:47.455
The determinant-- this should
be a capital Y right there--

00:14:47.455 --> 00:14:52.440
the determinant of Z is equal to
the sum, from j is equal to

00:14:52.440 --> 00:14:56.950
1 to n, of negative
1 to the i plus j.

00:14:56.950 --> 00:14:59.300
We're going along this row.

00:14:59.300 --> 00:15:06.160
But now the coefficients are xj,
that's what we're indexing

00:15:06.160 --> 00:15:09.820
along, xj plus yj.

00:15:09.820 --> 00:15:11.890
And then times its sub matrix,
which is the same

00:15:11.890 --> 00:15:13.582
as these sub matrices.

00:15:13.582 --> 00:15:20.500
So aij, which you might
immediately see is the sum of

00:15:20.500 --> 00:15:21.250
these two things.

00:15:21.250 --> 00:15:25.220
If I, for every j, I just summed
these two things up,

00:15:25.220 --> 00:15:27.650
you're having two coefficients--
you could have

00:15:27.650 --> 00:15:33.590
this coefficient and that
coefficient on your aij term,

00:15:33.590 --> 00:15:36.090
and then when you add them up
you can factor this guy out

00:15:36.090 --> 00:15:39.560
and you will get this
right here.

00:15:39.560 --> 00:15:42.730
So you get the determinant of X
plus the determinant of Y is

00:15:42.730 --> 00:15:46.470
equal to the determinant of Z.

00:15:46.470 --> 00:15:48.310
So hopefully that shows
you the general case.

00:15:48.310 --> 00:15:50.600
But I want to make it very
clear, this is just for a very

00:15:50.600 --> 00:15:55.140
particular scenario where three
matrices are identical,

00:15:55.140 --> 00:15:56.790
except for on one row.

00:15:56.790 --> 00:16:00.450
And one of the matrices on that
special row just happens

00:16:00.450 --> 00:16:04.600
to be the sum of the other two
matrices for that special row,

00:16:04.600 --> 00:16:06.320
and everything else
is identical.

00:16:06.320 --> 00:16:09.320
That's the only time where the
determinant of-- not the only

00:16:09.320 --> 00:16:10.930
time, but that's the only time
we can make the general

00:16:10.930 --> 00:16:13.145
statement where the determinant
of Z is equal to

00:16:13.145 --> 00:16:15.810
the determinant of X plus
the determinant of Y.

00:16:15.810 --> 00:16:19.340
It's not the case-- let me write
what is not the case--

00:16:19.340 --> 00:16:30.390
so not the case that if Z is
equal to X plus Y, it is not

00:16:30.390 --> 00:16:35.340
the case that the determinant
of Z is necessarily equal to

00:16:35.340 --> 00:16:43.450
the determinant of X plus
the determinant of Y.

00:16:43.450 --> 00:16:45.120
You cannot assume this.

00:16:45.120 --> 00:16:48.550
Determinant operations are not
linear on matrix addition.

00:16:48.550 --> 00:16:51.740
They're linear only on
particular rows getting at it.

00:16:51.740 --> 00:16:54.750
Anyway, hopefully you found
that vaguely useful.


(correction) scalar multiplication of row

WEBVTT
Kind: captions
Language: en

00:00:00.460 --> 00:00:03.260
I want to make a quick
correction or clarification to

00:00:03.260 --> 00:00:06.540
the last video that you may or
may not have found confusing.

00:00:06.540 --> 00:00:09.420
You may not have noticed it,
but when I did the general

00:00:09.420 --> 00:00:13.090
case for multiplying a row by a
scalar, I had this situation

00:00:13.090 --> 00:00:17.240
where I had the matrix A and I
defined it as-- it was n by n

00:00:17.240 --> 00:00:23.560
matrix, so it was a11, a12,
all the way to a1n.

00:00:23.560 --> 00:00:24.970
Then we went down this way.

00:00:24.970 --> 00:00:30.220
Then we picked a particular row
i, so we called that ai1,

00:00:30.220 --> 00:00:33.410
ai2, all the way to ain.

00:00:33.410 --> 00:00:36.065
And then we keep going down ,
assuming that this is the last

00:00:36.065 --> 00:00:40.250
row, so an1 all the
way to ann.

00:00:40.250 --> 00:00:42.500
When I wanted to find the
determinant of a, and this is

00:00:42.500 --> 00:00:46.770
where I made a-- I would call
it a notational error.

00:00:46.770 --> 00:00:51.360
When I wanted to find the
determinant of a, I wrote that

00:00:51.360 --> 00:00:55.460
it was equal to-- well, we could
go down, and in that

00:00:55.460 --> 00:00:57.010
video, I went down this row.

00:00:57.010 --> 00:00:59.350
That's why I kind of highlighted
it to begin with,

00:00:59.350 --> 00:01:00.770
and I wrote it down.

00:01:00.770 --> 00:01:03.370
So it's equal to-- do the
checkerboard pattern.

00:01:03.370 --> 00:01:06.610
I said negative 1
to the i plus j.

00:01:06.610 --> 00:01:07.640
Well, let's do the first term.

00:01:07.640 --> 00:01:16.240
I plus 1 times ai1 times
its submatrix.

00:01:16.240 --> 00:01:19.750
That's what I wrote in the last.
So if you have ai1, if

00:01:19.750 --> 00:01:22.810
you get rid of that row, that
column, you have the submatrix

00:01:22.810 --> 00:01:24.550
right there: ai1.

00:01:24.550 --> 00:01:26.550
That's what I wrote
in the last video,

00:01:26.550 --> 00:01:27.970
but that was incorrect.

00:01:27.970 --> 00:01:31.050
And I think when I did the 2 by
2 case and the 3 by 3 case,

00:01:31.050 --> 00:01:32.000
that's pretty clear.

00:01:32.000 --> 00:01:34.780
It's not times the matrix, it's
times the determinant of

00:01:34.780 --> 00:01:37.420
the submatrix, so this right
here is incorrect.

00:01:37.420 --> 00:01:40.770
And, of course, you keep adding
that to-- and I wrote

00:01:40.770 --> 00:01:44.520
ai2 times its submatrix
like that.

00:01:44.520 --> 00:01:50.620
ai2 all the way to ain
times its submatrix.

00:01:50.620 --> 00:01:51.560
That's what I did
in the video.

00:01:51.560 --> 00:01:52.780
That's incorrect.

00:01:52.780 --> 00:01:56.250
Let me do the incorrect in a
different color to show that

00:01:56.250 --> 00:01:57.680
this is all one thing.

00:01:57.680 --> 00:01:59.960
I should have said the
determinant of each of these.

00:01:59.960 --> 00:02:07.200
The determinant of a is equal
to minus 1 to the i plus 1

00:02:07.200 --> 00:02:16.180
times ai1 times the determinant
of ai1 plus ai2

00:02:16.180 --> 00:02:20.440
times the determinant of ai2,
the determinant of the

00:02:20.440 --> 00:02:26.440
submatrix all the way to ain
times the determinant of the

00:02:26.440 --> 00:02:29.440
submatrix ain.

00:02:29.440 --> 00:02:31.890
It doesn't change the logic of
the proof much, but I just

00:02:31.890 --> 00:02:33.910
want to be very careful that
we're not multiplying the

00:02:33.910 --> 00:02:35.850
submatrices because
that becomes a

00:02:35.850 --> 00:02:37.630
fairly complicated operation.

00:02:37.630 --> 00:02:38.260
Well, it's not that bad.

00:02:38.260 --> 00:02:38.820
It's a scalar.

00:02:38.820 --> 00:02:41.630
But when we find a determinant,
we're multiplying

00:02:41.630 --> 00:02:43.360
times the determinant
of the submatrix.

00:02:43.360 --> 00:02:45.850
We saw that when we first
defined it using the recursive

00:02:45.850 --> 00:02:48.760
definition for the n by n
determinant, but I just wanted

00:02:48.760 --> 00:02:51.230
to make that very clear.


Determinant when row multiplied by scalar

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:03.400
Let's explore what happens to
determinants when you multiply

00:00:03.400 --> 00:00:04.675
them by a scalar.

00:00:04.675 --> 00:00:08.210
So let's say we wanted to find
the determinant of this

00:00:08.210 --> 00:00:11.730
matrix, of a, b, c, d.

00:00:11.730 --> 00:00:15.820
By definition the determinant
here is going to be equal to a

00:00:15.820 --> 00:00:21.360
times d minus b times c, or
c times b, either way.

00:00:21.360 --> 00:00:22.810
ad minus bc.

00:00:22.810 --> 00:00:24.830
That's the determinant
right there.

00:00:24.830 --> 00:00:26.290
Now what if we were
to multiply one of

00:00:26.290 --> 00:00:27.590
these rows by a scalar.

00:00:27.590 --> 00:00:29.940
Let's say we multiply it by k.

00:00:29.940 --> 00:00:33.880
So we have the situation a, b.

00:00:33.880 --> 00:00:35.720
Let's multiply the second
row times k.

00:00:35.720 --> 00:00:39.870
So we kc and kd.

00:00:39.870 --> 00:00:42.350
Now what's the determinant
going to look like?

00:00:42.350 --> 00:00:44.840
We're going to have a times
kd, or we could just write

00:00:44.840 --> 00:00:52.410
that as kad, minus kc times b,
or we could right that as kbc.

00:00:52.410 --> 00:00:55.000
If we factor out the k,
we get that equals k

00:00:55.000 --> 00:00:58.690
times ad minus bc.

00:00:58.690 --> 00:01:02.000
Now you immediately see that
this thing is the same thing

00:01:02.000 --> 00:01:03.100
as this thing.

00:01:03.100 --> 00:01:10.930
So this is equal to k times the
determinant of a, b, c, d.

00:01:10.930 --> 00:01:13.910
So when you just multiply a
row by a scalar-- just one

00:01:13.910 --> 00:01:17.470
row, not the entire matrix--
when you just multiply a row

00:01:17.470 --> 00:01:20.770
by some scalar, the resulting
determinant will be the

00:01:20.770 --> 00:01:23.550
original determinant
times that scalar.

00:01:23.550 --> 00:01:26.880
Now you might say well what
happens if I multiply the

00:01:26.880 --> 00:01:28.580
whole matrix times
that scalar.

00:01:28.580 --> 00:01:33.040
Well that's equivalent to
multiplying by a scalar twice.

00:01:33.040 --> 00:01:36.120
Let's say I have the matrix A,
and the matrix A is equal to

00:01:36.120 --> 00:01:38.830
a, b, c, d.

00:01:38.830 --> 00:01:43.190
If I were to think about the
matrix kA, now I'm not just

00:01:43.190 --> 00:01:44.270
multiplying one row.

00:01:44.270 --> 00:01:46.210
I'm multiplying the whole
matrix by a scalar.

00:01:46.210 --> 00:01:55.500
This is going to be equal
to ka, kb, kc, and kd.

00:01:55.500 --> 00:02:00.180
When you figure out its
determinant, the determinant

00:02:00.180 --> 00:02:05.040
of k times A is going to be
equal to the determinant of

00:02:05.040 --> 00:02:13.880
ka, kb, kc, and kd.

00:02:13.880 --> 00:02:16.020
Here you're immediately going
to end up with k squared

00:02:16.020 --> 00:02:18.790
terms. You're going to have
k squared times ad.

00:02:18.790 --> 00:02:25.770
It's going to be equal to k
squared ad minus k squared bc,

00:02:25.770 --> 00:02:34.030
or k squared times ad minus
bc, or k squared times the

00:02:34.030 --> 00:02:37.080
determinant of just A.

00:02:37.080 --> 00:02:38.280
So you have to be
very careful.

00:02:38.280 --> 00:02:39.990
This is only for a
two by two case.

00:02:39.990 --> 00:02:43.140
You'll find out if this was an
n by n matrix that this would

00:02:43.140 --> 00:02:44.190
have been k to the n.

00:02:44.190 --> 00:02:46.600
So the take away is the only way
you can say that this is

00:02:46.600 --> 00:02:50.430
going to be some scalar multiple
times your original

00:02:50.430 --> 00:02:54.890
determinant, is only if you
multiply one row times that

00:02:54.890 --> 00:02:56.810
scalar multiple, not
the whole matrix.

00:02:56.810 --> 00:03:02.030
Let's see how this extends
to maybe a 3 by 3 case.

00:03:04.700 --> 00:03:06.370
You might say, hey Sal, you just
picked the second row.

00:03:06.370 --> 00:03:07.800
Does it work with
the first row?

00:03:07.800 --> 00:03:10.140
I'll leave that for you to
determine, but it does.

00:03:10.140 --> 00:03:10.940
It does work.

00:03:10.940 --> 00:03:13.500
It doesn't matter which row
I multiplied it by.

00:03:13.500 --> 00:03:16.450
Let's take the three
by three case.

00:03:16.450 --> 00:03:18.600
Let's say we have some matrix.

00:03:18.600 --> 00:03:20.020
Let's call this A again.

00:03:20.020 --> 00:03:21.600
I'm redefining A.

00:03:21.600 --> 00:03:29.300
It's going to be a, b,
c, d, e, f, g, h, i.

00:03:29.300 --> 00:03:35.370
Then if you take its
determinant, the determinant

00:03:35.370 --> 00:03:39.100
of A is going to be equal to--
we can do it a couple of

00:03:39.100 --> 00:03:39.560
different ways.

00:03:39.560 --> 00:03:41.870
But I'll just pick some
arbitrary row.

00:03:41.870 --> 00:03:43.670
Because that's the row that
we're going to multiply by

00:03:43.670 --> 00:03:44.410
some scalar.

00:03:44.410 --> 00:03:46.260
So let's just take that
row right there.

00:03:48.920 --> 00:03:50.590
Remember the plus
minus pattern.

00:03:50.590 --> 00:03:54.540
Remember plus, minus, plus,
minus, plus, minus, plus,

00:03:54.540 --> 00:03:56.630
minus, plus, that little
checkerboard pattern.

00:03:56.630 --> 00:03:58.590
So d is a minus right there.

00:03:58.590 --> 00:04:01.910
So it's going to be equal to
minus d times the determinant

00:04:01.910 --> 00:04:03.500
of its submatrix.

00:04:03.500 --> 00:04:05.340
So you cross out that
column in that row.

00:04:05.340 --> 00:04:07.020
It's b, c, h, i.

00:04:10.230 --> 00:04:15.986
It's going to be plus e times
its submatrix a, c, g, i.

00:04:19.810 --> 00:04:23.030
It's going to be minus f times--
you get rid of that

00:04:23.030 --> 00:04:24.970
row in that column--
d, e, g, h, the

00:04:24.970 --> 00:04:27.920
determinant of d, e, g, h.

00:04:27.920 --> 00:04:30.230
That's the determinant
of this matrix A.

00:04:30.230 --> 00:04:33.340
Now what if we define some
new matrix here?

00:04:33.340 --> 00:04:35.450
Let's call it A prime.

00:04:35.450 --> 00:04:38.180
Let me make this scroll
down a little bit.

00:04:38.180 --> 00:04:40.720
Let me define A prime
right here.

00:04:40.720 --> 00:04:45.010
A prime, I'm just going to
multiply this row by a scalar.

00:04:45.010 --> 00:04:52.590
So it's going to be equal to
a, b, c, kd, ke, and kf.

00:04:52.590 --> 00:04:54.640
I'm not multiplying the whole
matrix times the scalar.

00:04:54.640 --> 00:04:56.540
I can't say this is kA.

00:04:56.540 --> 00:04:58.540
I'm just multiplying
one of its rows.

00:04:58.540 --> 00:05:01.960
Then I have g, h, and i.

00:05:01.960 --> 00:05:06.400
So what's the determinant
of A prime going to be?

00:05:06.400 --> 00:05:07.340
I put that prime there.

00:05:07.340 --> 00:05:10.650
So it's different than A,
but it's derived from A.

00:05:10.650 --> 00:05:13.380
I just multiplied one row
of A times the scalar.

00:05:13.380 --> 00:05:16.810
Well I can go along that same
row that I did up here.

00:05:16.810 --> 00:05:18.580
I can go along that same row.

00:05:18.580 --> 00:05:21.230
The only difference is is that
instead of having a d,

00:05:21.230 --> 00:05:22.530
I now have a kd.

00:05:22.530 --> 00:05:23.870
Instead of an e, I
now have a ke.

00:05:23.870 --> 00:05:25.990
So instead of a d, I'm going
to have a kd there.

00:05:25.990 --> 00:05:28.300
Instead of an e, I'm going
to have an ke there.

00:05:28.300 --> 00:05:31.260
So it's going to be this exact
same thing, but I can replace

00:05:31.260 --> 00:05:35.160
this guy, this guy, and this guy
with them multiplied by k.

00:05:35.160 --> 00:05:40.070
So it's going to be equal to
minus kd times the determinant

00:05:40.070 --> 00:05:44.650
of the submatrix b, c, h, i--
I'm not even look over here

00:05:44.650 --> 00:05:46.420
because it's going to be the
same thing as that one up

00:05:46.420 --> 00:05:55.060
there-- plus ke times the
determinant of a, c, g, i,

00:05:55.060 --> 00:06:01.360
minus kf times the determinant
of d, e, g, h.

00:06:01.360 --> 00:06:02.800
What is this equal to?

00:06:02.800 --> 00:06:05.980
This is equal to, if you just
factor out the k, it's equal

00:06:05.980 --> 00:06:07.220
to k times this.

00:06:07.220 --> 00:06:10.680
So it's equal to k times
the determinant of A.

00:06:10.680 --> 00:06:13.150
So our result also worked
for three by three case.

00:06:13.150 --> 00:06:14.650
I just happened to pick
the middle row.

00:06:14.650 --> 00:06:16.780
But I encourage you
to pick other rows

00:06:16.780 --> 00:06:19.030
and to see what happens.

00:06:19.030 --> 00:06:21.470
So let's actually do it
for the general case.

00:06:21.470 --> 00:06:24.240
Because I've just been giving
you particular examples, and I

00:06:24.240 --> 00:06:26.500
like to show you the general
proof when the general proof

00:06:26.500 --> 00:06:27.920
isn't too harry.

00:06:27.920 --> 00:06:30.410
So let's say I have
an n by n matrix.

00:06:30.410 --> 00:06:32.670
Let's say that I
have matrix A.

00:06:32.670 --> 00:06:37.530
Let's say that A is n by n.

00:06:37.530 --> 00:06:38.940
You can write it like this.

00:06:38.940 --> 00:06:40.090
This is the first row.

00:06:40.090 --> 00:06:46.140
This is the first column a11,
a12, all the way to a1n.

00:06:46.140 --> 00:06:48.860
I'm going to pick some arbitrary
row here that I'm

00:06:48.860 --> 00:06:50.640
going to end up multiplying
by a scalar.

00:06:50.640 --> 00:06:52.550
So we can go down here.

00:06:52.550 --> 00:07:03.300
Let's say row ai, so this is
ai1, ai2, all the way to ain.

00:07:03.300 --> 00:07:05.610
This is some row that I'm going
to use to determine the

00:07:05.610 --> 00:07:06.060
determinant.

00:07:06.060 --> 00:07:08.600
Remember we can go to any row
to get the determinant.

00:07:08.600 --> 00:07:10.180
Then finally you keep going.

00:07:10.180 --> 00:07:18.860
You get an1, an2, all
the way to ann.

00:07:18.860 --> 00:07:23.860
This is as general as you can
get for an n by n matrix.

00:07:23.860 --> 00:07:27.000
Now let's figure out its
determinant, so the

00:07:27.000 --> 00:07:28.650
determinant of A.

00:07:28.650 --> 00:07:34.200
I'm just going to go down
this row right there.

00:07:34.200 --> 00:07:40.560
So the determinant of
A is equal to what?

00:07:40.560 --> 00:07:42.750
Well we have to remember the
checkerboard pattern.

00:07:42.750 --> 00:07:45.170
We don't know where we are on
the checkerboard pattern

00:07:45.170 --> 00:07:48.110
because I just picked an
arbitrary general row here.

00:07:48.110 --> 00:07:51.290
But we can use the general
formula that the sign is going

00:07:51.290 --> 00:07:56.270
to be determined by negative 1
to the i-- I don't know if i

00:07:56.270 --> 00:07:59.390
is even or odd-- so it's
going to be i plus

00:07:59.390 --> 00:08:01.110
for this term 1 power.

00:08:01.110 --> 00:08:01.920
That's it's sign.

00:08:01.920 --> 00:08:04.110
This is what gives us the
checkerboard pattern.

00:08:04.110 --> 00:08:05.190
Let me make that clear.

00:08:05.190 --> 00:08:07.260
It looks complicated, but this
is just a checkerboard.

00:08:13.510 --> 00:08:18.420
Times this term right there,
so times ai, so the

00:08:18.420 --> 00:08:23.720
coefficient, ai1, and then times
this guy's submatrix.

00:08:23.720 --> 00:08:24.730
You remember the submatrix.

00:08:24.730 --> 00:08:27.700
You get rid of this row, and
this column is going to be

00:08:27.700 --> 00:08:29.430
everything that's left over.

00:08:29.430 --> 00:08:32.960
So times that submatrix
of ai1.

00:08:35.539 --> 00:08:40.730
Then it's going to be plus-- we
just keep doing it-- plus

00:08:40.730 --> 00:08:46.360
negative 1 to the i plus
2 times ai2 times its

00:08:46.360 --> 00:08:53.300
submatrix-- all the way, you
just keep going-- plus minus 1

00:08:53.300 --> 00:08:56.380
to the i plus n times ai.

00:08:56.380 --> 00:08:59.170
You're in the nth column,
and then its submatrix.

00:08:59.170 --> 00:09:01.960
This is going to be an n minus
1 by n minus 1 matrix.

00:09:01.960 --> 00:09:03.990
All of these are going to be.

00:09:03.990 --> 00:09:06.520
Just like that, that's
the determinant of A.

00:09:06.520 --> 00:09:09.940
We could actually rewrite
it in sigma notation.

00:09:09.940 --> 00:09:11.890
That'll simplify things
a little bit.

00:09:11.890 --> 00:09:18.720
So the determinant of A, we can
rewrite it as the sum from

00:09:18.720 --> 00:09:23.600
j is equal to 1 to j-- I'll
write it explicitly here-- j

00:09:23.600 --> 00:09:36.800
is equal to n of negative 1 to
the i plus j times aij, and

00:09:36.800 --> 00:09:40.440
then each of the submatrices,
Aij.

00:09:40.440 --> 00:09:47.740
This thing right here is just
another way of writing this

00:09:47.740 --> 00:09:49.990
thing I wrote up there.

00:09:49.990 --> 00:09:50.790
I'm just saying the sum.

00:09:50.790 --> 00:09:52.500
You just take j equal 1,
put them in there.

00:09:52.500 --> 00:09:53.760
You get this term right there.

00:09:53.760 --> 00:09:55.830
You take j equal
2, you add it.

00:09:55.830 --> 00:09:56.660
You get this term right there.

00:09:56.660 --> 00:09:57.380
You keep doing it.

00:09:57.380 --> 00:09:58.480
You get j equal n.

00:09:58.480 --> 00:10:00.070
You get that term right there.

00:10:00.070 --> 00:10:02.800
So these two things
are equivalent.

00:10:02.800 --> 00:10:07.520
So what happens I have
some new matrix?

00:10:07.520 --> 00:10:11.370
Let me copy and paste
my current matrix.

00:10:11.370 --> 00:10:13.210
So let me copy and paste it.

00:10:13.210 --> 00:10:16.950
Actually let me copy and
paste everything.

00:10:16.950 --> 00:10:19.340
That'll make things
move quickly.

00:10:19.340 --> 00:10:20.410
I copied it.

00:10:20.410 --> 00:10:24.010
Now let me paste it
just like that.

00:10:24.010 --> 00:10:28.940
Let me define a new
matrix A prime.

00:10:28.940 --> 00:10:30.420
It's still an n by n matrix.

00:10:30.420 --> 00:10:33.240
But that row that I just
happened to use to determine

00:10:33.240 --> 00:10:36.500
my determinant, I'm going to
multiply it by a scalar k.

00:10:36.500 --> 00:10:43.740
So it's k ai1, k ai2 k
ain just like that.

00:10:43.740 --> 00:10:46.580
So what's the determinate
of A prime?

00:10:46.580 --> 00:10:49.090
Well we're just going to
go down this row again.

00:10:49.090 --> 00:10:53.200
But now instead of just an ai1,
we have a k ai1 Instead

00:10:53.200 --> 00:10:55.970
of an ai2, we have a
k ai2 Instead of an

00:10:55.970 --> 00:10:58.530
ain we have a k ain.

00:10:58.530 --> 00:11:01.830
So it's determinant is just
going to be this same thing,

00:11:01.830 --> 00:11:04.375
but instead of an aij
everywhere, we're going to

00:11:04.375 --> 00:11:06.540
have a k aij.

00:11:06.540 --> 00:11:08.203
So this is the determinant
of A prime.

00:11:11.710 --> 00:11:15.180
We could just take out this
constant right here.

00:11:15.180 --> 00:11:17.100
It has no i's or j's in it.

00:11:17.100 --> 00:11:19.310
I has no j's in it in
particular, so we can just

00:11:19.310 --> 00:11:20.270
take it out.

00:11:20.270 --> 00:11:26.100
So it's equal to k times the sum
from j is equal to 1 to j

00:11:26.100 --> 00:11:34.080
is equal to n of minus 1 to
the i plus j times aij.

00:11:34.080 --> 00:11:35.710
This is the coefficient.

00:11:35.710 --> 00:11:39.040
This is the submatrix for each
of those coefficients, aij.

00:11:39.040 --> 00:11:41.170
That's a matrix right
there, an n minus 1

00:11:41.170 --> 00:11:43.310
by n minus 1 matrix.

00:11:43.310 --> 00:11:45.550
Then you immediately recognize--
I think you saw

00:11:45.550 --> 00:11:48.580
where this was going-- this
right here is just the

00:11:48.580 --> 00:11:52.820
determinant of A.

00:11:52.820 --> 00:11:57.430
So we get the result that the
determinant of A prime is

00:11:57.430 --> 00:12:02.890
equal to k times the
determinant of A.

00:12:02.890 --> 00:12:05.535
So we've just shown you in
general, if you have any n by

00:12:05.535 --> 00:12:10.680
n matrix, if you multiply only
one row, not the whole matrix,

00:12:10.680 --> 00:12:15.900
only one row by some scalar
multiple k, the resulting

00:12:15.900 --> 00:12:19.440
determinant will be your
original determinant times k.

00:12:19.440 --> 00:12:22.060
Now I touched on this in
the original video.

00:12:26.820 --> 00:12:30.810
What is the determinant
of k times A?

00:12:30.810 --> 00:12:34.400
So now we're multiplying
every row times k.

00:12:34.400 --> 00:12:38.940
Or another way to think about
is you're multiplying

00:12:38.940 --> 00:12:41.500
n rows times k.

00:12:41.500 --> 00:12:43.810
So you're doing this n times.

00:12:43.810 --> 00:12:45.880
So if you multiply
k times itself n

00:12:45.880 --> 00:12:46.930
times, what do you get?

00:12:46.930 --> 00:12:49.010
You get k to the n.

00:12:49.010 --> 00:12:53.090
So this is going to be equal
to k to the n times the

00:12:53.090 --> 00:12:55.560
determinant of A.

00:12:55.560 --> 00:12:57.750
If you just do it once,
you get k times the

00:12:57.750 --> 00:12:58.460
determinant of A.

00:12:58.460 --> 00:13:00.650
Now if you do a second row,
you're going to get k times k

00:13:00.650 --> 00:13:01.715
times the determinant of A.

00:13:01.715 --> 00:13:03.730
If you do a third row, you're
going to get k to the third

00:13:03.730 --> 00:13:05.010
times the determinant of A.

00:13:05.010 --> 00:13:06.040
The fourth row, k to the
fourth times the

00:13:06.040 --> 00:13:06.900
determinant of A.

00:13:06.900 --> 00:13:09.386
If you do them all, all n rows,
you're going to have k

00:13:09.386 --> 00:13:11.480
to the n times the
determinant of A.

00:13:11.480 --> 00:13:14.140
Anyway, hopefully you found
that interesting.

00:13:14.140 --> 00:13:15.970
I encourage you to experiment
with these other ways.

00:13:15.970 --> 00:13:19.390
Try going down a column and
seeing what happens.


Scientific notation examples

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.620
It always helps me to see a lot
of examples of something so I

00:00:03.620 --> 00:00:05.920
figured it wouldn't hurt to
do more scientific

00:00:05.920 --> 00:00:06.980
notation examples.

00:00:06.980 --> 00:00:09.920
So I'm just going to write a
bunch of numbers and then write

00:00:09.920 --> 00:00:11.390
them in scientific notation.

00:00:11.390 --> 00:00:14.180
And hopefully this'll cover
almost every case you'll ever

00:00:14.180 --> 00:00:16.160
see and then at the end of this
video, we'll actually do some

00:00:16.160 --> 00:00:18.430
computation with them to just
make sure that we can

00:00:18.430 --> 00:00:20.960
do computation with
scientific notation.

00:00:20.960 --> 00:00:24.835
Let me just write down
a bunch of numbers.

00:00:24.835 --> 00:00:28.380
0.00852.

00:00:28.380 --> 00:00:29.640
That's my first number.

00:00:29.640 --> 00:00:39.445
My second number
is 7012000000000.

00:00:39.445 --> 00:00:42.040
I'm just arbitrarily
stopping the zeroes.

00:00:42.040 --> 00:00:50.620
The next number is 0.0000000
I'll just draw a couple more.

00:00:50.620 --> 00:00:55.640
If I keep saying 0, you
might find that annoying.

00:00:55.640 --> 00:01:01.260
500 The next number --
right here, there's a

00:01:01.260 --> 00:01:02.580
decimal right there.

00:01:02.580 --> 00:01:09.090
The next number I'm going
to do is the number 723.

00:01:09.090 --> 00:01:11.970
The next number I'll do -- I'm
having a lot of 7's here.

00:01:11.970 --> 00:01:13.260
Let's do 0.6.

00:01:16.300 --> 00:01:19.980
And then let's just do one more
just for, just to make sure

00:01:19.980 --> 00:01:21.590
we've covered all of our bases.

00:01:21.590 --> 00:01:26.640
Let's say we do 823 and then
let's throw some -- an

00:01:26.640 --> 00:01:29.640
arbitrary number of 0's there.

00:01:29.640 --> 00:01:33.460
So this first one, right here,
what we do if we want to write

00:01:33.460 --> 00:01:35.910
in scientific notation, we want
to figure out the largest

00:01:35.910 --> 00:01:38.100
exponent of 10 that
fits into it.

00:01:38.100 --> 00:01:40.310
So we go to its first
non-zero term, which

00:01:40.310 --> 00:01:41.230
is that right there.

00:01:41.230 --> 00:01:45.250
We count how many positions to
the right of the decimal point

00:01:45.250 --> 00:01:47.000
we have including that term.

00:01:47.000 --> 00:01:49.070
So we have one, two, three.

00:01:49.070 --> 00:01:51.560
So it's going to
be equal to this.

00:01:51.560 --> 00:01:54.230
So it's going to be equal
to 8 -- that's that guy

00:01:54.230 --> 00:01:56.370
right there -- 0.52.

00:01:56.370 --> 00:01:58.220
So everything after that
first term is going to

00:01:58.220 --> 00:01:59.190
be behind the decimal.

00:01:59.190 --> 00:02:03.340
So 0.52 times 10 to the
number of terms we have.

00:02:03.340 --> 00:02:04.520
One, two, three.

00:02:04.520 --> 00:02:07.390
10 to the minus 3.

00:02:07.390 --> 00:02:09.440
Another way to think of it:
this is a little bit more.

00:02:09.440 --> 00:02:12.290
This is like 8 1/2
thousands, right?

00:02:12.290 --> 00:02:13.430
Each of these is thousands.

00:02:13.430 --> 00:02:15.290
We have 8 1/2 of them.

00:02:15.290 --> 00:02:16.520
Let's do this one.

00:02:16.520 --> 00:02:17.900
Let's see how many 0's we have.

00:02:17.900 --> 00:02:23.620
We have 3, 6, 9, 12.

00:02:23.620 --> 00:02:26.630
So we want to do -- again,
we start with our largest

00:02:26.630 --> 00:02:27.730
term that we have.

00:02:27.730 --> 00:02:29.335
Our largest non-zero term.

00:02:29.335 --> 00:02:30.750
In this case, it's going
to be the term all

00:02:30.750 --> 00:02:31.430
the way to the left.

00:02:31.430 --> 00:02:32.900
That's our 7.

00:02:32.900 --> 00:02:35.570
So it's going to be 7.012.

00:02:35.570 --> 00:02:40.960
It's going to be equal to
7.012 times 10 to the what?

00:02:40.960 --> 00:02:45.350
Well it's going to be times 10
to the 1 with this many 0's.

00:02:45.350 --> 00:02:46.300
So how many things?

00:02:46.300 --> 00:02:48.300
We had a 1 here.

00:02:48.300 --> 00:02:56.700
Then we had 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12 0's.

00:02:56.700 --> 00:02:57.490
I want to be very clear.

00:02:57.490 --> 00:02:58.600
You're not just
counting the 0's.

00:02:58.600 --> 00:03:02.860
You're counting everything
after this first

00:03:02.860 --> 00:03:03.730
term right there.

00:03:03.730 --> 00:03:07.550
So it would be equivalent
to a 1 followed by 12 0's.

00:03:07.550 --> 00:03:11.360
So it's times 10
to the twelfth.

00:03:11.360 --> 00:03:12.020
Just like that.

00:03:12.020 --> 00:03:12.790
Not too difficult.

00:03:12.790 --> 00:03:14.630
Let's do this one right here.

00:03:14.630 --> 00:03:16.650
So we go behind our
decimal point.

00:03:16.650 --> 00:03:18.520
We find the first
non-zero number.

00:03:18.520 --> 00:03:19.980
That's our 5.

00:03:19.980 --> 00:03:21.610
It's going to be equal to 5.

00:03:21.610 --> 00:03:24.800
There's nothing to the right of
it, so it's 5.00 if we wanted

00:03:24.800 --> 00:03:26.710
to add some precision to it.

00:03:26.710 --> 00:03:30.880
But it's 5 times and then how
many numbers to the right, or

00:03:30.880 --> 00:03:32.790
behind to the right of the
decimal will do we have?

00:03:32.790 --> 00:03:40.760
We have 1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12, 13, and we

00:03:40.760 --> 00:03:42.820
have to include this one, 14.

00:03:42.820 --> 00:03:47.700
5 times 10 to the
minus 14th power.

00:03:47.700 --> 00:03:50.640
Now this number, it might be a
little overkill to write this

00:03:50.640 --> 00:03:52.670
in scientific notation, but it
never hurts to get

00:03:52.670 --> 00:03:53.700
the practice.

00:03:53.700 --> 00:03:56.490
So what's the largest 10
that goes into this?

00:03:56.490 --> 00:03:58.640
Well, 100 will go into this.

00:03:58.640 --> 00:04:02.150
And you could figure out 100 or
10 squared by saying, "OK, this

00:04:02.150 --> 00:04:06.990
is our largest term." And then
we have two 0's behind it

00:04:06.990 --> 00:04:11.210
because we can say 100
will go into 723.

00:04:11.210 --> 00:04:17.070
So this is going to be equal to
7.23 times, we could say times

00:04:17.070 --> 00:04:19.995
100, but we want to stay in
scientific notation, so I'll

00:04:19.995 --> 00:04:22.240
write times 10 squared.

00:04:22.240 --> 00:04:24.360
Now we have this
character right here.

00:04:24.360 --> 00:04:25.960
What's our first non-zero term?

00:04:25.960 --> 00:04:29.310
It's that one right there, so
it's going to be 6 times and

00:04:29.310 --> 00:04:31.300
then how many terms do we have
to the right of the decimal?

00:04:31.300 --> 00:04:32.350
We have only one.

00:04:32.350 --> 00:04:34.300
So times 10 to the minus 1.

00:04:34.300 --> 00:04:36.530
That makes a lot of sense
because that's essentially

00:04:36.530 --> 00:04:39.170
equal to 6 divided by 10
because 10 to the minus

00:04:39.170 --> 00:04:42.510
1 is 1/10 which is 0.6.

00:04:42.510 --> 00:04:44.120
One more.

00:04:44.120 --> 00:04:45.770
Let me throw some commas
here just to make this a

00:04:45.770 --> 00:04:48.530
little easier to look at.

00:04:48.530 --> 00:04:50.610
So let's take our largest
value right there.

00:04:50.610 --> 00:04:53.790
We have our 8.

00:04:53.790 --> 00:04:59.220
This is going to be 8.23 -- we
don't have to add the other

00:04:59.220 --> 00:05:03.250
stuff because everything else
is a 0 -- times 10 to the --

00:05:03.250 --> 00:05:06.070
we just count how many
terms are after the 8.

00:05:06.070 --> 00:05:12.960
So we have 1, 2, 3, 4,
5, 6, 7, 8, 9, 10.

00:05:12.960 --> 00:05:15.170
8.23 times 10 to the 10.

00:05:15.170 --> 00:05:16.290
I think you get the idea now.

00:05:16.290 --> 00:05:17.590
It's pretty straightforward.

00:05:17.590 --> 00:05:20.860
And more than just being able
to calculate this, which is a

00:05:20.860 --> 00:05:23.200
good skill by itself, I want
you to understand why

00:05:23.200 --> 00:05:23.940
this is the case.

00:05:23.940 --> 00:05:25.820
Hopefully that last
video explained it.

00:05:25.820 --> 00:05:28.390
And if it doesn't, just
multiply this out.

00:05:28.390 --> 00:05:31.430
Literally multiply 8.23
times 10 to the 10 and

00:05:31.430 --> 00:05:32.770
you will get this number.

00:05:32.770 --> 00:05:34.190
Maybe you could try it
with something smaller

00:05:34.190 --> 00:05:35.040
than 10 to the 10.

00:05:35.040 --> 00:05:36.110
Maybe 10 to the fifth.

00:05:36.110 --> 00:05:38.140
And well, you'll get a
different number but

00:05:38.140 --> 00:05:40.980
you'll end up with five
digits after the 8.

00:05:40.980 --> 00:05:45.180
But anyway, let me do a couple
more computation examples.

00:05:45.180 --> 00:05:54.790
Let's say we had the numbers
-- let me just make something

00:05:54.790 --> 00:05:58.410
really small -- 0.0000064.

00:05:58.410 --> 00:06:00.060
Let me make a large number.

00:06:00.060 --> 00:06:03.100
Let's say I have that number
and I want to multiply it.

00:06:03.100 --> 00:06:06.450
I want to multiply it by --
let's say I have a really large

00:06:06.450 --> 00:06:11.610
number -- 3 2 -- I'm just going
to throw a bunch of 0's here.

00:06:11.610 --> 00:06:12.530
I don't know when
I'm going to stop.

00:06:12.530 --> 00:06:13.900
Let's say I stop there.

00:06:13.900 --> 00:06:15.560
So this one, you
can multiply out.

00:06:15.560 --> 00:06:18.650
But it's a little difficult.

00:06:18.650 --> 00:06:20.680
But let's put it into
scientific notation.

00:06:20.680 --> 00:06:23.090
One, it'll be easier to
represent these numbers and

00:06:23.090 --> 00:06:25.800
then hopefully you'll see that
the multiplication actually

00:06:25.800 --> 00:06:27.720
gets simplified as well.

00:06:27.720 --> 00:06:30.550
So this top guy right here,
how can we write him in

00:06:30.550 --> 00:06:31.110
scientific notation?

00:06:31.110 --> 00:06:37.560
It would be 6.4 times
10 to the what?

00:06:37.560 --> 00:06:40.430
1, 2, 3, 4, 5, 6.

00:06:40.430 --> 00:06:41.440
I have to include the 6.

00:06:41.440 --> 00:06:43.480
So times 10 to the minus 6.

00:06:43.480 --> 00:06:45.800
And what can this
one be written as?

00:06:45.800 --> 00:06:47.930
This one is going to be 3.2.

00:06:51.150 --> 00:06:53.560
And then you count how many
digits are after the 3.

00:06:53.560 --> 00:06:59.330
1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11.

00:06:59.330 --> 00:07:02.750
So 3.2 times 10 to the 11th.

00:07:02.750 --> 00:07:06.187
So if we multiply these two
things, this is equivalent to 6

00:07:06.187 --> 00:07:12.650
-- let me do it in a different
color -- 6.4 times 10 to

00:07:12.650 --> 00:07:21.590
the minus 6 times 3.2
times 10 to the 11th.

00:07:21.590 --> 00:07:26.160
Which we saw in the last video
is equivalent to 6.4 times 3.2.

00:07:26.160 --> 00:07:29.360
I'm just changing the order
of our multiplication.

00:07:29.360 --> 00:07:36.560
Times 10 to the minus 6
times 10 to the 11th power.

00:07:36.560 --> 00:07:38.300
And now what will
this be equal to?

00:07:38.300 --> 00:07:40.490
Well, to do this, I don't
want to use a calculator.

00:07:40.490 --> 00:07:42.530
So let's just calculate it.

00:07:42.530 --> 00:07:47.830
So 6.4 times 3.2.

00:07:47.830 --> 00:07:49.450
Let's ignore the
decimals for a second.

00:07:49.450 --> 00:07:50.640
We'll worry about
that at the end.

00:07:50.640 --> 00:07:55.060
So 2 times 4 is 8,
2 times 6 is 12.

00:07:55.060 --> 00:07:57.770
Nowhere to carry the
1, so it's just 128.

00:07:57.770 --> 00:07:58.810
Put a 0 down there.

00:07:58.810 --> 00:08:02.290
3 times 4 is 12, carry the 1.

00:08:02.290 --> 00:08:04.370
3 times 6 is 18.

00:08:04.370 --> 00:08:08.320
You've got a 1
there, so it's 192.

00:08:08.320 --> 00:08:08.700
Right?

00:08:08.700 --> 00:08:09.510
Yeah.

00:08:09.510 --> 00:08:09.670
192.

00:08:09.670 --> 00:08:13.920
You had them up and you
get 8, 4, 1 plus 9 is 10.

00:08:13.920 --> 00:08:14.590
Carry the 1.

00:08:14.590 --> 00:08:15.780
You get 2.

00:08:15.780 --> 00:08:17.920
Now, we just have to
count the numbers behind

00:08:17.920 --> 00:08:19.130
the decimal point.

00:08:19.130 --> 00:08:21.280
We have one number there, we
have another number there.

00:08:21.280 --> 00:08:23.045
We have two numbers behind
the decimal point,

00:08:23.045 --> 00:08:25.120
so you count 1, 2.

00:08:25.120 --> 00:08:34.880
So 6.4 times 3.2 is equal to
20.48 times 10 to the -- we

00:08:34.880 --> 00:08:37.930
have the same base here, so we
can just add the exponents.

00:08:37.930 --> 00:08:40.450
So what's minus 6 plus 11?

00:08:40.450 --> 00:08:45.830
So that's 10 to the
fifth power, right?

00:08:45.830 --> 00:08:46.180
Right.

00:08:46.180 --> 00:08:47.580
Minus 6 and 11.

00:08:47.580 --> 00:08:49.040
10 to the fifth power.

00:08:49.040 --> 00:08:50.690
And so the next question,
you might say, "I'm done.

00:08:50.690 --> 00:08:52.940
I've done the computation."
And you have.

00:08:52.940 --> 00:08:55.030
And this is a valid answer.

00:08:55.030 --> 00:08:58.270
But the next question is is
this in scientific notation?

00:08:58.270 --> 00:09:01.180
And if you wanted to be a real
stickler about it, it's not in

00:09:01.180 --> 00:09:03.530
scientific notation because we
have something here that could

00:09:03.530 --> 00:09:06.140
maybe be simplified
a little bit.

00:09:06.140 --> 00:09:09.100
We could write this --
let me do it this way.

00:09:09.100 --> 00:09:10.960
Let me divide this by 10.

00:09:10.960 --> 00:09:13.950
So any number we can
multiply and divide by 10.

00:09:13.950 --> 00:09:16.130
So we could rewrite
it this way.

00:09:16.130 --> 00:09:19.610
We could write 1/10 on this
side and then we can multiply

00:09:19.610 --> 00:09:21.270
times 10 on that side, right?

00:09:21.270 --> 00:09:22.910
That shouldn't
change the number.

00:09:22.910 --> 00:09:24.650
You divide by 10 and
multiply it by 10.

00:09:24.650 --> 00:09:27.770
That's just like multiplying
by 1 or dividing by 1.

00:09:27.770 --> 00:09:33.010
So if you divide this side
by 10, you get 2.048.

00:09:33.010 --> 00:09:36.280
You multiply that side by
10 and you get times 10 to

00:09:36.280 --> 00:09:38.730
the -- times 10 is just
times 10 to the first.

00:09:38.730 --> 00:09:39.780
You can just add the exponents.

00:09:39.780 --> 00:09:41.200
Times 10 to the sixth.

00:09:41.200 --> 00:09:44.280
And now, if you're a stickler
about it, this is good

00:09:44.280 --> 00:09:48.840
scientific notation
right there.

00:09:48.840 --> 00:09:50.620
Now, I've done a lot
of multiplication.

00:09:50.620 --> 00:09:54.510
Let's do some division.

00:09:54.510 --> 00:09:57.010
Let's divide this
guy by that guy.

00:09:57.010 --> 00:10:04.980
So if we have 3.2 times 10 to
the eleventh power divided by

00:10:04.980 --> 00:10:09.600
6.4 times 10 to the minus
six, what is this equal to?

00:10:09.600 --> 00:10:13.940
Well, this is equal
to 3.2 over 6.4.

00:10:13.940 --> 00:10:16.300
We can just separate them out
because it's associative.

00:10:16.300 --> 00:10:23.210
So, it's this times 10
to the 11th over 10 to

00:10:23.210 --> 00:10:24.290
the minus six, right?

00:10:24.290 --> 00:10:25.530
If you multiply these
two things, you'll

00:10:25.530 --> 00:10:26.930
get that right there.

00:10:26.930 --> 00:10:30.130
So 3.2 over 6.4.

00:10:30.130 --> 00:10:32.030
This is just equal
to 0.5, right?

00:10:32.030 --> 00:10:36.170
32 is half of 64 or 3.2
is half of 6.4, so this

00:10:36.170 --> 00:10:38.140
is 0.5 right there.

00:10:38.140 --> 00:10:39.350
And what is this?

00:10:39.350 --> 00:10:43.100
This is 10 to the 11th
over 10 to the minus 6.

00:10:43.100 --> 00:10:45.700
So when you have something
in the denominator, you

00:10:45.700 --> 00:10:46.620
could write it this way.

00:10:46.620 --> 00:10:51.510
This is equivalent to 10 to the
11th over 10 to the minus 6.

00:10:51.510 --> 00:10:56.070
It's equal to 10 to the
11th times 10 to the

00:10:56.070 --> 00:10:58.800
minus 6 to the minus 1.

00:10:58.800 --> 00:11:03.060
Or this is equal to 10 to the
11th times 10 to the sixth.

00:11:03.060 --> 00:11:04.610
And what did I do just there?

00:11:04.610 --> 00:11:07.140
This is 1 over 10
to the minus 6.

00:11:07.140 --> 00:11:09.580
So 1 over something is just
that something to the

00:11:09.580 --> 00:11:10.500
negative 1 power.

00:11:10.500 --> 00:11:12.280
And then I multiplied
the exponents.

00:11:12.280 --> 00:11:14.110
You can think of it that way
and so this would be equal

00:11:14.110 --> 00:11:19.000
to 10 to the 17th power.

00:11:19.000 --> 00:11:21.620
Or another way to think about
it is if you have 1 -- you have

00:11:21.620 --> 00:11:26.360
the same bases, 10 in this
case, and you're dividing them,

00:11:26.360 --> 00:11:29.400
you just take the 1 the
numerator and you subtract the

00:11:29.400 --> 00:11:30.220
exponent in the denominator.

00:11:30.220 --> 00:11:35.010
So it's 11 minus minus
6, which is 11 plus 6,

00:11:35.010 --> 00:11:36.910
which is equal to 17.

00:11:36.910 --> 00:11:41.210
So this division problem
ended up being equal to

00:11:41.210 --> 00:11:46.290
0.5 times 10 to the 17th.

00:11:46.290 --> 00:11:49.150
Which is the correct answer,
but if you wanted to be a

00:11:49.150 --> 00:11:51.410
stickler and put it into
scientific notation, we want

00:11:51.410 --> 00:11:54.190
something maybe greater
than 1 right here.

00:11:54.190 --> 00:11:55.800
So the way we can do
that, let's multiply

00:11:55.800 --> 00:11:59.420
it by 10 on this side.

00:11:59.420 --> 00:12:02.990
And divide by 10 on this
side or multiply by 1/10.

00:12:02.990 --> 00:12:04.880
Remember, we're not changing
the number if you multiply

00:12:04.880 --> 00:12:06.660
by 10 and divide by 10.

00:12:06.660 --> 00:12:08.950
We're just doing it to
different parts of the product.

00:12:08.950 --> 00:12:15.630
So this side is going to become
5 -- I'll do it in pink -- 10

00:12:15.630 --> 00:12:20.930
times 0.5 is 5, times 10 to
the 17th divided by 10.

00:12:20.930 --> 00:12:24.190
That's the same thing as
10 to the 17th times 10

00:12:24.190 --> 00:12:25.650
to the minus 1, right?

00:12:25.650 --> 00:12:27.120
That's 10 to the minus 1.

00:12:27.120 --> 00:12:29.320
So it's equal to 10
to the 16th power.

00:12:31.930 --> 00:12:35.330
Which is the answer when
you divide these two

00:12:35.330 --> 00:12:36.490
guys right there.

00:12:36.490 --> 00:12:39.750
So hopefully these examples
have filled in all of the

00:12:39.750 --> 00:12:42.250
gaps or the uncertain
scenarios dealing with

00:12:42.250 --> 00:12:43.260
scientific notation.

00:12:43.260 --> 00:12:46.220
If I haven't covered something,
feel free to write a comment on

00:12:46.220 --> 00:12:48.510
this video or pop me an e-mail.


Rule of Sarrus of determinants

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:02.930
I don't want to beat a dead
horse by showing you all of

00:00:02.930 --> 00:00:05.610
the different ways to find a
determinant, but it might be

00:00:05.610 --> 00:00:09.110
useful to beat this dead horse
because you'll see it done in

00:00:09.110 --> 00:00:10.830
different ways, in different
context.

00:00:10.830 --> 00:00:13.230
And I thought I would at least
show you that what we have

00:00:13.230 --> 00:00:16.190
been covering, so far, is very
consistent with a way of

00:00:16.190 --> 00:00:18.880
determining, or finding
determinants, that you might

00:00:18.880 --> 00:00:21.370
have been exposed to in your
Algebra Two class.

00:00:21.370 --> 00:00:24.250
It's called the rule
of Sarrus.

00:00:24.250 --> 00:00:25.470
Let me just prove it for you.

00:00:25.470 --> 00:00:27.520
Let's say we want to
find a determinant.

00:00:27.520 --> 00:00:28.850
Let's say you want to find
this determinant.

00:00:28.850 --> 00:00:35.870
So our matrix is a, b,
c, d, e, f, g, h, i.

00:00:35.870 --> 00:00:36.980
We know how to do this.

00:00:36.980 --> 00:00:40.180
This is equal to, let's just
go down that first row, a

00:00:40.180 --> 00:00:49.710
times the determinant of e,
f, h, i minus b times the

00:00:49.710 --> 00:00:57.830
determinant of d, g, f,
i plus c times the

00:00:57.830 --> 00:01:00.065
determinant of d, e, g, h.

00:01:03.320 --> 00:01:04.989
And what are these equal to?

00:01:04.989 --> 00:01:09.160
This is going to be equal to
a, so let me write this, a

00:01:09.160 --> 00:01:14.460
times ei minus fh.

00:01:17.150 --> 00:01:24.645
And this is going to be minus
b times dI minus fg.

00:01:24.645 --> 00:01:33.160
This is going to be plus c times
dh minus eg and if we

00:01:33.160 --> 00:01:41.550
multiplied this out we get this
is equal to aei minus afh

00:01:41.550 --> 00:01:51.610
minuses bdi plus right, minus
times a minus, plus bfg plus

00:01:51.610 --> 00:01:56.120
cdh minus ceg.

00:01:56.120 --> 00:01:58.840
Now let me group the positive
and the negative terms.

00:01:58.840 --> 00:02:02.980
So this term is positive, this
term is positive and that term

00:02:02.980 --> 00:02:03.600
is positive.

00:02:03.600 --> 00:02:12.410
So we have this being equal
to aei plus bfg plus cdh.

00:02:12.410 --> 00:02:13.900
Those are our positive terms.

00:02:13.900 --> 00:02:16.640
And then our negative
terms are here.

00:02:16.640 --> 00:02:19.250
We have that term, that
term, and that term.

00:02:19.250 --> 00:02:29.450
So we have minus afh minus
bdi minus ceg.

00:02:29.450 --> 00:02:33.320
So this is a formula for the
determinant of this matrix

00:02:33.320 --> 00:02:33.800
right here.

00:02:33.800 --> 00:02:35.170
Let's see what it actually
looks like.

00:02:35.170 --> 00:02:36.570
Let me rewrite it.

00:02:36.570 --> 00:02:39.570
Let me rewrite our matrix.

00:02:39.570 --> 00:02:41.790
We do it in green.

00:02:41.790 --> 00:02:50.560
So we have a, b, c,
d, e, f, g, h, i.

00:02:50.560 --> 00:02:53.230
We wanted to find
its determinant.

00:02:53.230 --> 00:02:55.590
So let me show you something
interesting here.

00:02:55.590 --> 00:02:57.750
aei is what?

00:02:57.750 --> 00:03:00.930
aei is a product of this guy,
this guy, and that guy.

00:03:00.930 --> 00:03:03.270
So, you're essentially
going along that

00:03:03.270 --> 00:03:05.340
diagonal right there.

00:03:05.340 --> 00:03:08.490
Now what is bfg?

00:03:08.490 --> 00:03:10.600
You're going this guy, this guy,
and then you're going all

00:03:10.600 --> 00:03:11.650
the way down to this guy.

00:03:11.650 --> 00:03:14.320
So it's like if you imagine that
when you come out of this

00:03:14.320 --> 00:03:15.490
side, you come out
of this side.

00:03:15.490 --> 00:03:17.830
There's some video games where
you go out one end and you end

00:03:17.830 --> 00:03:19.610
up showing up on the other
end like that.

00:03:19.610 --> 00:03:20.770
It would also be a diagonal.

00:03:20.770 --> 00:03:25.070
Or even a better way to
visualize it, let me redraw

00:03:25.070 --> 00:03:26.080
these two columns.

00:03:26.080 --> 00:03:28.810
Let me augment this
determinant.

00:03:28.810 --> 00:03:31.770
It's not official terminology,
but I think you'll get what

00:03:31.770 --> 00:03:33.410
I'm trying to do.

00:03:33.410 --> 00:03:37.060
So if I write these first
two columns again. a, d,

00:03:37.060 --> 00:03:41.430
g, and b, e, h.

00:03:41.430 --> 00:03:45.780
This guy right here bfg, it's
this one right here, this

00:03:45.780 --> 00:03:47.420
diagonal right there.

00:03:47.420 --> 00:03:49.800
And then you might guess
what's about to happen.

00:03:49.800 --> 00:03:51.290
Where is cdh?

00:03:51.290 --> 00:03:52.990
It's this diagonal.

00:03:52.990 --> 00:03:55.210
It's that diagonal
right there.

00:03:55.210 --> 00:03:57.800
So you take this product, add it
to this product, add it to

00:03:57.800 --> 00:03:58.770
this product.

00:03:58.770 --> 00:04:00.330
And then you subtract
these guys.

00:04:00.330 --> 00:04:02.320
Now what are these guys?

00:04:02.320 --> 00:04:03.570
Where is the afh?

00:04:05.710 --> 00:04:06.830
That one right there.

00:04:06.830 --> 00:04:11.770
So you subtract out afh, and
then you subtract out bdi.

00:04:11.770 --> 00:04:16.380
bdi is that one right there.

00:04:16.380 --> 00:04:23.220
And then you have ceg, which
is this one right there.

00:04:23.220 --> 00:04:27.330
So the Rule of Sarrus, sounds
like something in

00:04:27.330 --> 00:04:28.510
The Lord of the Rings.

00:04:28.510 --> 00:04:37.560
The Rule of Sarrus is
essentially a quick way of

00:04:37.560 --> 00:04:39.450
memorizing this little
technique.

00:04:39.450 --> 00:04:42.560
You write the two columns
again, you say, ok, this

00:04:42.560 --> 00:04:45.620
product plus this product plus
this product, minus this

00:04:45.620 --> 00:04:48.220
product minus this product
minus that product.

00:04:48.220 --> 00:04:51.625
Let's actually do it with the 3
by 3 matrix to make it clear

00:04:51.625 --> 00:04:55.800
that the Rule of Sarrus
can be useful.

00:04:55.800 --> 00:04:57.990
So let's say we have the
matrix, we want the

00:04:57.990 --> 00:05:08.710
determinant of the matrix, 1, 2,
4, 2, minus 1, 3, and then

00:05:08.710 --> 00:05:12.740
we have 4, 0, minus 1.

00:05:12.740 --> 00:05:14.720
We want to find that
determinant.

00:05:14.720 --> 00:05:17.550
So by the Rule of Sarrus,
we can rewrite

00:05:17.550 --> 00:05:18.830
these first two columns.

00:05:18.830 --> 00:05:22.360
So 1, 2, 2, minus 1, 4, 0.

00:05:22.360 --> 00:05:24.610
We rewrote those first
two columns.

00:05:24.610 --> 00:05:29.950
And to figure out this
determinant we take this guy.

00:05:29.950 --> 00:05:30.670
What is this going to be?

00:05:30.670 --> 00:05:33.410
1 times minus 1 times minus 1.

00:05:33.410 --> 00:05:35.280
That is just a 1.

00:05:35.280 --> 00:05:37.220
Right, the minuses cancel out.

00:05:37.220 --> 00:05:40.320
Plus this guy, plus this
product right here.

00:05:40.320 --> 00:05:41.890
I should draw a little
bit neater.

00:05:41.890 --> 00:05:42.380
So what is this?

00:05:42.380 --> 00:05:44.130
2 times 3 times 4.

00:05:44.130 --> 00:05:46.500
2 two times 3 is 6.

00:05:46.500 --> 00:05:49.680
6 times 4 is 24, plus 24.

00:05:49.680 --> 00:05:51.480
And then we take this
guy right here.

00:05:51.480 --> 00:05:55.110
4 times 2 times 0, anything
times 0 is a 0.

00:05:55.110 --> 00:05:56.570
So that's going to be plus 0.

00:05:56.570 --> 00:05:59.810
And then we subtract
out these guys.

00:05:59.810 --> 00:06:04.780
So you have 4 times
4, times minus 1.

00:06:04.780 --> 00:06:06.276
That's minus 16.

00:06:06.276 --> 00:06:09.700
It's minus 16, but we're
going to be on the

00:06:09.700 --> 00:06:11.350
minus side of things.

00:06:11.350 --> 00:06:14.300
So it's 4 times minus 1
times 4, is minus 16.

00:06:14.300 --> 00:06:16.340
But since were going to
do a minus on it, it's

00:06:16.340 --> 00:06:18.310
going to be plus 16.

00:06:18.310 --> 00:06:19.660
So it's 16.

00:06:19.660 --> 00:06:23.230
Then you have a 0
times 3 times 1.

00:06:23.230 --> 00:06:24.510
That of course is
going to be 0.

00:06:24.510 --> 00:06:26.350
Would be a minus 0, but
we can ignore it.

00:06:26.350 --> 00:06:29.270
So we can say plus 0 or
minus 0 same thing.

00:06:29.270 --> 00:06:37.020
Then you have a minus
1 times 2 times 2.

00:06:37.020 --> 00:06:39.880
So that's 4 times minus
1 which is minus 4.

00:06:39.880 --> 00:06:42.050
When you go in this direction,
from the top right to the

00:06:42.050 --> 00:06:43.780
bottom left, you are
subtracting.

00:06:43.780 --> 00:06:46.380
So this would be a minus 4 but
since we're subtracting, this

00:06:46.380 --> 00:06:48.530
becomes a plus 4.

00:06:48.530 --> 00:06:52.390
So the value of our determinant
is equal to, by

00:06:52.390 --> 00:06:56.230
the Rule of Sarrus, we're
going to have

00:06:56.230 --> 00:07:00.370
16 plus 4 is a 20.

00:07:00.370 --> 00:07:05.960
20 plus 25 which
is equal to 45.

00:07:05.960 --> 00:07:10.120
So that actually is, I'd have
to say, a faster way of

00:07:10.120 --> 00:07:12.580
computing this 3 by
3 derivative.

00:07:12.580 --> 00:07:14.330
And I just want to show you this
is completely equivalent

00:07:14.330 --> 00:07:16.200
to the definition that I
introduced you to a couple

00:07:16.200 --> 00:07:17.450
videos ago.


Determinants along other rows/cols

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:03.920
In the last video, we evaluated
this 4 by 4

00:00:03.920 --> 00:00:07.400
determinant and we found out
that it was equal to 7.

00:00:07.400 --> 00:00:10.230
And the way we did it is we
went down this first row.

00:00:10.230 --> 00:00:12.870
We used the definition I gave
you in the last few where use

00:00:12.870 --> 00:00:13.980
this first row.

00:00:13.980 --> 00:00:15.000
I could even write it here.

00:00:15.000 --> 00:00:17.890
We said this is equal
to 1 times the

00:00:17.890 --> 00:00:20.660
determinant of 0, 2, 0.

00:00:20.660 --> 00:00:22.520
1, 2, 3.

00:00:22.520 --> 00:00:24.440
3, 0, 0.

00:00:24.440 --> 00:00:26.950
Minus 2 times the determinant.

00:00:26.950 --> 00:00:28.440
You cross these guys.

00:00:28.440 --> 00:00:29.980
Cross that row and column out.

00:00:29.980 --> 00:00:31.310
1, 0, 2.

00:00:31.310 --> 00:00:32.970
1, 0, 2.

00:00:32.970 --> 00:00:34.750
2, 2, 0.

00:00:34.750 --> 00:00:36.770
0, 3, 0.

00:00:36.770 --> 00:00:40.180
And then we went to the plus
the 3 times its sub matrix.

00:00:40.180 --> 00:00:41.240
I don't have to figure
that out.

00:00:41.240 --> 00:00:43.150
Just cross out that row
and that column.

00:00:43.150 --> 00:00:47.950
And then minus 4-- just keep
switching the sign-- times the

00:00:47.950 --> 00:00:49.570
determinant of its sub matrix.

00:00:49.570 --> 00:00:51.410
So this one had a bunch of
terms. And this one's going to

00:00:51.410 --> 00:00:53.710
have a bunch of terms. Cross
that row and that column out.

00:00:53.710 --> 00:00:54.710
You get 1, 0, 2.

00:00:54.710 --> 00:00:55.930
0, 1, 2.

00:00:55.930 --> 00:00:57.000
2, 3, 0.

00:00:57.000 --> 00:00:57.710
I'll just write it here.

00:00:57.710 --> 00:00:59.600
1, 0, 2.

00:00:59.600 --> 00:01:01.130
0, 1, 2.

00:01:01.130 --> 00:01:02.770
2, 3, 0.

00:01:02.770 --> 00:01:05.610
And that is a completely
legitimate way to figure out a

00:01:05.610 --> 00:01:06.110
determinant.

00:01:06.110 --> 00:01:08.600
And that was our definition for
how to find a determinant.

00:01:08.600 --> 00:01:10.890
But I want to show you in this
video that there's more than

00:01:10.890 --> 00:01:12.970
one way to solve for
determinant.

00:01:12.970 --> 00:01:14.550
What I'm going to show you this
way is the same thing

00:01:14.550 --> 00:01:16.560
that we did down
this first row.

00:01:16.560 --> 00:01:20.000
We can actually do down any
row or any column of this

00:01:20.000 --> 00:01:22.610
determinant, or of
this matrix.

00:01:22.610 --> 00:01:25.010
And the reason why that's useful
is because we can pick

00:01:25.010 --> 00:01:28.040
rows or columns that
have a unusually

00:01:28.040 --> 00:01:29.490
large number of zeros.

00:01:29.490 --> 00:01:32.350
Because that tends to simplify
our computations.

00:01:32.350 --> 00:01:35.100
So the first thing that you have
to do before you embark

00:01:35.100 --> 00:01:38.070
on picking an arbitrary row or
a column, let's say, for

00:01:38.070 --> 00:01:40.320
example, we want to pick this.

00:01:40.320 --> 00:01:42.390
Let's do one row and one
column in this example.

00:01:42.390 --> 00:01:44.580
So let's say we want to go down
that row instead because

00:01:44.580 --> 00:01:47.150
we like the fact that has
a lot of zeros there.

00:01:47.150 --> 00:01:49.620
The first thing you have to do
is remember the pattern.

00:01:49.620 --> 00:01:52.210
Remember, you switch signs
on the coefficients.

00:01:52.210 --> 00:01:54.350
You don't just switch signs
as you go down a row.

00:01:54.350 --> 00:01:56.930
You also switch signs as
you go down a column.

00:01:56.930 --> 00:02:00.710
So the general pattern for 4
by 4 will look like this.

00:02:00.710 --> 00:02:05.450
It'll be plus, minus,
plus, minus, minus,

00:02:05.450 --> 00:02:08.380
plus, minus, plus.

00:02:08.380 --> 00:02:12.090
And then you get a plus, minus,
minus, plus, plus,

00:02:12.090 --> 00:02:13.950
minus, minus, plus.

00:02:13.950 --> 00:02:15.870
It's really this checker
board pattern.

00:02:15.870 --> 00:02:20.200
If you wanted to figure out
the sign for any ij.

00:02:20.200 --> 00:02:22.020
So let's say you wanted
to figure out the sign

00:02:22.020 --> 00:02:24.170
for, this is 2, 2.

00:02:24.170 --> 00:02:27.360
So if you want to find
the sign-- let me

00:02:27.360 --> 00:02:28.370
write it this way.

00:02:28.370 --> 00:02:31.670
Say we have a function, let
me define a function.

00:02:31.670 --> 00:02:33.570
I think the checker board
pattern's pretty clear to you,

00:02:33.570 --> 00:02:34.470
but I'll just write it down.

00:02:34.470 --> 00:02:38.760
Let's say I wanted to figure
out the sign-- the sign not

00:02:38.760 --> 00:02:42.280
the trig ratio.

00:02:42.280 --> 00:02:46.820
I want to figure out the sign of
any entry where you give me

00:02:46.820 --> 00:02:48.440
an i and a j.

00:02:48.440 --> 00:02:50.620
What you could do is you
just take negative 1 to

00:02:50.620 --> 00:02:53.070
the i plus j power.

00:02:53.070 --> 00:02:57.190
So if you wanted to figure out
the sign for the, let's say,

00:02:57.190 --> 00:03:01.720
you are in row 4, column 2.

00:03:01.720 --> 00:03:04.040
Row 4, column 2.

00:03:04.040 --> 00:03:04.890
So what will it be?

00:03:04.890 --> 00:03:07.550
Do 4 plus 2.

00:03:07.550 --> 00:03:09.990
Negative 1 to the sixth
power is equal to 1.

00:03:09.990 --> 00:03:12.850
That's going to be
a positive 1.

00:03:16.650 --> 00:03:17.840
Let's say you want
to take this guy.

00:03:17.840 --> 00:03:22.860
So this is i is equal to 2.

00:03:22.860 --> 00:03:24.350
j is equal to 3.

00:03:24.350 --> 00:03:26.580
We're in the second
row, third column.

00:03:26.580 --> 00:03:28.150
2 plus 3 is 5.

00:03:28.150 --> 00:03:31.160
Minus 1 to the fifth
power is minus 1.

00:03:31.160 --> 00:03:32.280
And you have a minus there.

00:03:32.280 --> 00:03:34.290
So that's another way
to think about it.

00:03:34.290 --> 00:03:35.640
But the checker board
pattern's pretty

00:03:35.640 --> 00:03:36.650
straightforward.

00:03:36.650 --> 00:03:38.530
So now that you have the checker
board pattern in your

00:03:38.530 --> 00:03:40.860
mind, let's go down this row.

00:03:40.860 --> 00:03:42.770
Let's go down this row.

00:03:42.770 --> 00:03:44.260
So we start with the 2.

00:03:44.260 --> 00:03:46.630
But notice that we have
to multiply by minus.

00:03:46.630 --> 00:03:50.090
Because we go plus, minus,
plus, minus.

00:03:50.090 --> 00:03:52.610
So you have a minus 2 times the

00:03:52.610 --> 00:03:54.480
determinant of its sub matrix.

00:03:54.480 --> 00:03:56.350
So you cross out this
row and that column.

00:03:56.350 --> 00:03:58.530
And you're left with this
matrix up here.

00:03:58.530 --> 00:04:01.490
So it's 2, 3, 4.

00:04:01.490 --> 00:04:04.050
0, 2, 0.

00:04:04.050 --> 00:04:07.650
1, 2, 3.

00:04:07.650 --> 00:04:09.960
This is a minus and then
you have a plus.

00:04:09.960 --> 00:04:13.880
So you have a plus 3 times--
get rid of this

00:04:13.880 --> 00:04:15.370
guy's column and row.

00:04:15.370 --> 00:04:16.750
And you have 1, 1, 0.

00:04:16.750 --> 00:04:18.190
1, 1, 0.

00:04:18.190 --> 00:04:19.360
That's that right there.

00:04:19.360 --> 00:04:20.970
3, 2, 2.

00:04:20.970 --> 00:04:22.850
3, 2, 2.

00:04:22.850 --> 00:04:24.940
And you have 4, 0, 3.

00:04:24.940 --> 00:04:27.020
4, 0, 3.

00:04:27.020 --> 00:04:29.630
And then you would have a
minus a 0 times its sub

00:04:29.630 --> 00:04:30.900
matrix plus a 0.

00:04:30.900 --> 00:04:32.560
But we can ignore those
because 0 times

00:04:32.560 --> 00:04:33.910
anything is a 0.

00:04:33.910 --> 00:04:38.230
So already we've simplified our
determinant a good bit.

00:04:38.230 --> 00:04:39.950
So let's see if we can
evaluate this and

00:04:39.950 --> 00:04:41.170
get the same number.

00:04:41.170 --> 00:04:43.810
Because only then will it be
reasonably satisfying.

00:04:43.810 --> 00:04:45.410
So what's the determinant
of this guy?

00:04:45.410 --> 00:04:47.360
Well we can do the exact
same principle.

00:04:47.360 --> 00:04:49.810
We can go down any row or
column that seems to be

00:04:49.810 --> 00:04:51.240
especially simple.

00:04:51.240 --> 00:04:54.370
So let's go down that row.

00:04:54.370 --> 00:04:56.230
Because that row seems
especially simple.

00:04:56.230 --> 00:04:58.470
So this is going
to be minus 2.

00:04:58.470 --> 00:05:01.010
That's this minus 2 right
here times the

00:05:01.010 --> 00:05:02.330
determinant of this guy.

00:05:02.330 --> 00:05:05.220
So the determinant of this guy,
we just have to go and

00:05:05.220 --> 00:05:08.120
say OK we have a plus,
we have a minus, and

00:05:08.120 --> 00:05:08.720
then we have a plus.

00:05:08.720 --> 00:05:13.192
It's going to be minus 0 times
its sub determinant, I guess

00:05:13.192 --> 00:05:13.740
we would call it.

00:05:13.740 --> 00:05:17.240
Get rid of that row or that
column and that row.

00:05:17.240 --> 00:05:18.910
So it'd be minus 0 times
anything, that's just going to

00:05:18.910 --> 00:05:20.850
be 0 plus 2.

00:05:20.850 --> 00:05:24.560
So plus 2 times the
determinant.

00:05:24.560 --> 00:05:26.610
Get rid of its row
and its columns.

00:05:26.610 --> 00:05:28.140
2, 4, 1, 2.

00:05:28.140 --> 00:05:32.280
2, 4, 1, 3.

00:05:32.280 --> 00:05:34.830
And then you have a minus
0 times this thing.

00:05:34.830 --> 00:05:37.500
But who cares what that is
because you have a 0 times.

00:05:37.500 --> 00:05:40.630
So that just simplified
to that which is nice.

00:05:40.630 --> 00:05:43.000
Let me write it like that.

00:05:43.000 --> 00:05:49.940
And then you have plus 3 times
this thing right here.

00:05:49.940 --> 00:05:50.950
We don't want to do
the first row.

00:05:50.950 --> 00:05:52.520
We have no non 0 terms here.

00:05:52.520 --> 00:05:55.910
Let's at least do this
row right here for a

00:05:55.910 --> 00:05:58.130
little bit of variety.

00:05:58.130 --> 00:06:00.300
None of the columns actually
seem that interesting.

00:06:00.300 --> 00:06:03.010
They all have, at most, one 0.

00:06:03.010 --> 00:06:06.730
So if we do that one right here,
this is a plus, minus,

00:06:06.730 --> 00:06:09.270
plus, minus, plus.

00:06:09.270 --> 00:06:12.300
So we'll have a plus
0 times 3, 4, 2, 0.

00:06:12.300 --> 00:06:13.420
We can ignore that.

00:06:13.420 --> 00:06:15.900
Minus 2 times the determinant.

00:06:15.900 --> 00:06:19.410
Get rid of this column,
that row.

00:06:19.410 --> 00:06:20.420
I have to be very careful.

00:06:20.420 --> 00:06:23.880
I put this minus there, but
there wasn't a minus 1 there.

00:06:23.880 --> 00:06:29.240
Let me write it like-- I want to
make sure I don't make any

00:06:29.240 --> 00:06:30.490
careless errors right here.

00:06:30.490 --> 00:06:31.970
This is a plus 1.

00:06:31.970 --> 00:06:34.070
I just drew a minus 1 there
to show you how

00:06:34.070 --> 00:06:35.530
things switch signs.

00:06:35.530 --> 00:06:40.070
So this is going to
be a 3 times.

00:06:40.070 --> 00:06:43.060
So we're going to go down--
This is this 3 right here.

00:06:43.060 --> 00:06:45.090
I lost my bearings with
that minus there.

00:06:45.090 --> 00:06:46.950
We're trying to find the
determinant of this.

00:06:46.950 --> 00:06:49.340
So it's 0 times this matrix.

00:06:49.340 --> 00:06:50.650
We can ignore that.

00:06:50.650 --> 00:06:53.320
Minus 2 times its sub matrix.

00:06:53.320 --> 00:06:55.330
Which is that and that.

00:06:55.330 --> 00:06:59.320
So it's 1, 1, 4, 0.

00:06:59.320 --> 00:07:02.240
And then you have a plus
3 times its sub matrix.

00:07:02.240 --> 00:07:03.630
1, 3, 1, 2.

00:07:03.630 --> 00:07:06.070
1, 3, 1, 2.

00:07:06.070 --> 00:07:08.020
Just like that.

00:07:08.020 --> 00:07:10.130
Let's see if we can
simplify this.

00:07:10.130 --> 00:07:14.040
2 times 3 is 6 minus
1 times 4.

00:07:14.040 --> 00:07:16.220
So, this becomes 6 minus 4.

00:07:16.220 --> 00:07:17.350
So that's 2.

00:07:17.350 --> 00:07:20.780
So this whole thing simplifies
to 2 times 2 which is 4.

00:07:20.780 --> 00:07:22.100
4 times minus 2.

00:07:22.100 --> 00:07:26.470
The whole thing simplifies
to minus 8.

00:07:26.470 --> 00:07:31.230
Now, this guy right here, we
have 1 times 0, which is 0,

00:07:31.230 --> 00:07:32.730
minus 1 times 4.

00:07:32.730 --> 00:07:35.610
This is minus 4 times minus 2.

00:07:35.610 --> 00:07:37.870
This whole thing becomes
a positive 8.

00:07:37.870 --> 00:07:41.840
You have 1 times 2, which
is 2, minus 1 times 3.

00:07:41.840 --> 00:07:44.260
Which is minus 3.

00:07:44.260 --> 00:07:45.140
So you get a minus 1.

00:07:45.140 --> 00:07:49.220
So this becomes a minus--
I mean you get 2 minus 3

00:07:49.220 --> 00:07:50.450
minus 1 times 3.

00:07:50.450 --> 00:07:52.960
So this becomes a minus 3.

00:07:52.960 --> 00:07:55.220
So you have an 8 minus a 3.

00:07:55.220 --> 00:07:57.333
So this becomes a minus five.

00:08:01.930 --> 00:08:05.050
You have a 3 times a minus
5 right there.

00:08:05.050 --> 00:08:12.700
So 3 times minus 5 is going
to be equal to minus 15.

00:08:12.700 --> 00:08:16.530
Let me make sure I got-- oh
I made a silly mistake.

00:08:16.530 --> 00:08:19.730
If you have an 8 minus a 3, this
is 8, this is minus, this

00:08:19.730 --> 00:08:21.340
is going to be 5.

00:08:21.340 --> 00:08:21.900
Very easy.

00:08:21.900 --> 00:08:24.760
Your brain starts to get fried
if you do this long enough.

00:08:24.760 --> 00:08:27.510
And the you have a 3 times
a 5 and you get a 15.

00:08:27.510 --> 00:08:28.650
You get a 15.

00:08:28.650 --> 00:08:30.030
And then you have a 15.

00:08:30.030 --> 00:08:35.169
This term plus this term is 15
minus 8 which is equal to 7.

00:08:35.169 --> 00:08:39.500
Which, lucky for us, I barely
evaded making a careless

00:08:39.500 --> 00:08:41.250
mistake there, we got
the right answer.

00:08:41.250 --> 00:08:44.190
But this is a much simpler
computation than we did in the

00:08:44.190 --> 00:08:45.140
last video.

00:08:45.140 --> 00:08:48.450
And it was much simpler because
we picked the row that

00:08:48.450 --> 00:08:50.880
happened to have a lot
of zeros on it.

00:08:50.880 --> 00:08:53.590
So we only had two of these
terms instead of four terms

00:08:53.590 --> 00:08:55.170
like we had in the last video.

00:08:55.170 --> 00:08:57.770
And you could do the same, you
could pick columns that have a

00:08:57.770 --> 00:08:58.590
lot of zeros.

00:08:58.590 --> 00:09:01.020
You just have to make sure
that you always use this

00:09:01.020 --> 00:09:02.900
checker board pattern.