WEBVTT
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ANA RITA PIRES: Hi.
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Welcome back to recitation.
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In lecture, you've been
learning about the properties
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of determinants.
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To remember, there were
three main properties,
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and then seven more that
fall out of those three.
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I'll tell you what
these three were.
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The first one was the
determinant of the identity
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matrix is always equal to 1.
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If you switch two
rows in a matrix,
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the determinant switches sign.
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And the determinant
is a function
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of each-- it's a linear
function of each row separately.
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And there's seven more.
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We'll use them here.
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Today's problem is about finding
the determinants of matrices
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by using these properties.
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So here you have four matrices.
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A has lots of 100's,
200's, and 300's numbers.
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B is called a
Vandermonde matrix.
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It has a very nice structure
with 1's, and then a, b, c;
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a squared, b squared, c squared.
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It can be bigger,
and you'll just
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have cubes and more
letters down here.
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C is given by the
product of these two,
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and D is this matrix.
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Good luck.
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Hit pause, work on them, and
when you're ready, come back
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and I'll show you how I did it.
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Did you get some?
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OK, let's do it.
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Starting with matrix A. I
have lots of big numbers.
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I suggest that we do a
little bit of elimination,
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because as you know,
doing elimination steps,
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except for permuting
rows, doesn't change
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the determinant of the matrix.
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So let's do determinant of A
is equal to-- 101, 201, 301.
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Then if I subtract off the
first row from the second one,
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I'll get 1, 1, 1, which
is very convenient.
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And actually, if you
subtract the second row
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from the third one,
you'll get 1, 1, 1.
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Here's the property
of the determinant:
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if you have two equal
rows on your matrix,
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the determinant is
automatically equal to zero.
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All right.
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All done with one of them.
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Let's work on the second one.
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The determinant of B. Well,
let's try elimination again.
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1, a, a squared.
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1 minus 1 is 0, b minus a,
and b squared minus a squared.
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b squared minus a squared--
let me factor that
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into b minus a, b plus
a, which will be very
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convenient in the next step.
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And then I'm going to
subtract the first row again
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from the third one.
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I'll get 0, c minus a.
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And again, I'll get c
squared minus a squared, c
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minus a, c plus a.
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Let's use that third property
that was the determinant
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is linear on each
row separately.
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So what I'm going to do is,
see this factor of b minus a?
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It shows up in every
entry of this row.
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Well it's a 0, so it's the
zero multiple of b minus a.
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So I'm going to pull out
this factor of b minus a,
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and this row is going to
become 0, 1, b plus a.
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I will also, in the same
step, do the same thing
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with the third row.
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I'll pull out a
factor of c minus a,
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and it will become
0, 1, c plus a.
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Here's one factor
from the second row.
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Another factor
from the third row.
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And 1, a, a squared.
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0, 1, b plus a.
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0, 1, c plus a.
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Now what?
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Well remember,
you know how to do
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the determinant of upper
triangular matrices
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because all that you do
is multiply the pivots.
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This is almost upper triangular,
except there's a 1 over here.
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So let's do another
elimination step.
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b minus a, c minus a.
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1, a, a squared; 0, 1, b
plus a; 0, 0, c plus a minus
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b plus a is c minus b.
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So the determinant of this
matrix is now 1 times 1 times
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c minus b.
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So we get b minus a,
c minus a, c minus b,
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which has a really nice formula.
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This is called the
Vandermonde determinant.
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It's always like this, even if
your matrix is bigger than 3
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by 3, if it's 4 by 4, or
5 by 5 and so on, you just
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have more differences
of all the letters
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that show up in your matrix.
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On to matrix number 3.
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C equals [1, 2, 3] [1, -4, 5].
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How did you get the
determinant for this one?
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Well remember, this
is a rank one matrix
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because it's a column
vector times a row vector.
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So if you write out what
the matrix is it'll be a 3
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by 3 matrix where we can
think about it this way.
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The first row will be
1 times [1, -4, 5].
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The second row will be
2 times those numbers.
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And the third row will be
three times those numbers.
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So all the rows are going
to be linearly dependent,
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or another way of saying
the matrix is singular.
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When the matrix is
singular, the determinant
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is always equal to 0.
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That's also one
of the properties.
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C is 0.
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Onto the next matrix.
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Last one.
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D is equal to 0, 0, 0.
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1, -1, 3, -3, 4, -4.
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Maybe you didn't get that from
just looking at the matrix
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the first time, but did you
see how I wrote it down?
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It has 0's down the diagonal.
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And then for each entry,
I have minus that entry.
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That means that this
matrix is skew symmetric.
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What that means is if
you do D transpose,
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it will not be
equal to D, but it
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will be equal to minus D. D
transpose is equal to minus D.
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Now, what does this give
us for a determinant?
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Well if these two matrices
are the same matrix,
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this determinant is equal
to that determinant.
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One of the properties
is that the determinant
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of a transpose of the matrix
is equal to the determinant
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of the original matrix.
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How about this side?
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Well, first temptation
would be to just write that.
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Is that always true?
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No.
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The determinant is linear
on each row separately.
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That means that you can't
pull out the factor that
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is multiplying the matrix.
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You have to pull it out once
for each row of the matrix.
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So what I should
have written was
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-1, that's my factor, minus.
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How many rows do I have?
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One, two, three.
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Pull it out once for each row.
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Times the determinant of D. Well
fortunately, -1 to the third
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is simply equal to -1.
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So here we go.
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It was correct, in fact.
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We have determinant of D is
equal to minus determinant
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of D. What is the
only number that
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is equal to minus that number?
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0.
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Determinant of D is
equal to 0 again.
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Let me ask you
one last question.
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Is it true that all skew
symmetric matrices have
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the determinant equal to 0?
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It was true for this one.
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Is it true in every case?
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Well, the key factor
here was that I
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had -1 to the third power
and I got a minus sign here,
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determinant of D is equal
to minus determinant of D.
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What if this number had
been an even number?
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Then I would just have
the determinant of D
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is equal minus 1 to an
even number, D. That's 1.
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So I would have determinant of
D is equal to determinant of D.
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There's nothing I can
say about that number.
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It can be your favorite
number, not necessarily 0.
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All right.
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We're done for today.
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Thank you.