WEBVTT
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I've been multiplying
matrices already,
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but certainly time for
me to discuss the rules
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for matrix multiplication.
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And the interesting part is
the many ways you can do it,
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and they all give
the same answer.
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And they're all important.
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So matrix multiplication,
and then, come inverses.
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So we mentioned the
inverse of a matrix.
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That's a big deal.
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Lots to do about inverses
and how to find them.
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Okay, so I'll begin with how
to multiply two matrices.
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First way, okay, so suppose
I have a matrix A multiplying
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a matrix B and --
giving me a result --
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well, I could call it C.
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A times B.
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Okay.
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So, let me just review
the rule for this entry.
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That's the entry in
row i and column j.
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So that's the i j entry.
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Right there is C i j.
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We always write the row number
and then the column number.
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So I might --
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I might -- maybe I take it C
3 4, just to make it specific.
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So instead of i j,
let me use numbers.
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C 3 4.
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So where does that come
from, the three four entry?
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It comes from row three, here,
row three and column four,
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as you know.
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Column four.
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And can I just write
down, or can we
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write down the formula for it?
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If we look at the whole
row and the whole column,
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the quick way for me to
say it is row three of A --
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I could use a dot
for dot product.
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I won't often use
that, actually.
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Dot column four of B.
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But this gives us a
chance to just, like,
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use a little matrix notation.
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What are the entries?
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What's this first
entry in row three?
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That number that's
sitting right there is...
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A, so it's got two
indices and what are they?
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3 1.
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So there's an a 3 1 there.
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Now what's the first guy
at the top of column four?
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So what's sitting up there?
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B 1 4, right.
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So that this dot product
starts with A 3 1 times B 1 4.
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And then what's the next -- so
this is like I'm accumulating
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this sum, then comes the next
guy, A 3 2, second column,
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times B 2 4, second row.
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So it's b A 3 2,
B 2 4 and so on.
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Just practice with indices.
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Oh, let me even practice
with a summation formula.
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So this is --
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most of the course,
I use whole vectors.
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I very seldom, get
down to the details
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of these particular entries,
but here we'd better do it.
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So it's some kind
of a sum, right?
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Of things in row three,
column K shall I say?
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Times things in
row K, column four.
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Do you see that that's
what we're seeing here?
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This is K is one, here
K is two, on along --
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so the sum goes all the
way along the row and down
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the column, say, one to N.
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So that's what the C three
four entry looks like.
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A sum of a three K b K four.
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Just takes a little
practice to do that.
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Okay.
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And -- well, maybe
I should say --
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when are we allowed to
multiply these matrices?
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What are the shapes
of these things?
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The shapes are --
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if we allow them to be not
necessarily square matrices.
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If they're square,
they've got to be the same
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size.
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If they're rectangular,
they're not the same size.
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If they're rectangular,
this might be -- well,
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I always think of A as m by n.
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m rows, n columns.
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So that sum goes to n.
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Now what's the point -- how
many rows does B have to have?
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n.
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The number of rows in
B, the number of guys
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that we meet coming down has
to match the number of ones
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across.
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So B will have to
be n by something.
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Whatever.
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P. So the number of columns here
has to match the number of rows
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there, and then
what's the result?
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What's the shape of the result?
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What's the shape
of C, the output?
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Well, it's got these same
m rows -- it's got m rows.
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And how many columns?
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P. m by P. Okay.
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So there are m times P little
numbers in there, entries,
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and each one, looks like that.
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Okay.
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So that's the standard rule.
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That's the way people think
of multiplying matrices.
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I do it too.
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But I want to talk
about other ways
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to look at that
same calculation,
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looking at whole
columns and whole rows.
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Okay.
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So can I do A B C again?
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A B equaling C again?
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But now, tell me about...
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I'll put it up here.
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So here goes A, again,
times B producing C.
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And again, this is m by n.
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This is n by P and
this is m by P. Okay.
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Now I want to look
at whole columns.
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I want to look at
the columns of --
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here's the second way
to multiply matrices.
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Because I'm going to build
on what I know already.
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How do I multiply a
matrix by a column?
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I know how to multiply
this matrix by that column.
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Shall I call that column one?
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That tells me column
one of the answer.
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The matrix times the first
column is that first column.
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Because none of
this stuff entered
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that part of the answer.
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The matrix times
the second column
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is the second column
of the answer.
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Do you see what I'm saying?
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That I could think of
multiplying a matrix
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by a vector, which I
already knew how to do,
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and I can think of just P
columns sitting side by side,
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just like resting
next to each other.
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And I multiply A times
each one of those.
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And I get the P
columns of the answer.
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Do you see this as --
this is quite nice,
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to be able to think, okay,
matrix multiplication works
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so that I can just think
of having several columns,
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multiplying by A and getting
the columns of the answer.
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So, like, here's column one
shall I call that column one?
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And what's going in there
is A times column one.
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Okay.
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So that's the picture
a column at a time.
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So what does that tell me?
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What does that tell me
about these columns?
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These columns of C
are combinations,
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because we've seen that
before, of columns of A.
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Every one of these
comes from A times this,
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and A times a vector
is a combination
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of the columns of A.
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And it makes sense, because
the columns of A have length m
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and the columns of
C have length m.
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And every column of
C is some combination
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of the columns of A.
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And it's these
numbers in here that
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tell me what combination it is.
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Do you see that?
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That in that answer, C,
I'm seeing stuff that's
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combinations of these columns.
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Now, suppose I look at it
-- that's two ways now.
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The third way is
look at it by rows.
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So now let me change to rows.
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Okay.
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So now I can think
of a row of A --
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a row of A multiplying all these
rows here and producing a row
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of the product.
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So this row takes a
combination of these rows
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and that's the answer.
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So these rows of C are
combinations of what?
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Tell me how to finish that.
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The rows of C, when I have a
matrix B, it's got its rows
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and I multiply by A,
and what does that do?
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It mixes the rows up.
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It creates combinations
of the rows of B, thanks.
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Rows of B.
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That's what I wanted to
see, that this answer --
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I can see where the
pieces are coming from.
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The rows in the answer are
coming as combinations of these
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rows.
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The columns in the answer are
coming as combinations of those
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columns.
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And so that's three ways.
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Now you can say, okay,
what's the fourth way?
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The fourth way --
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so that's -- now we've
got, like, the regular way,
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the column way,
the row way and --
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what's left?
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The one that I can --
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well, one way is
columns times rows.
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What happens if I multiply --
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So this was row times
column, it gave a number.
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Okay.
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Now I want to ask you
about column times row.
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If I multiply a column
of A times a row of B,
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what shape I ending up with?
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So if I take a
column times a row,
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that's definitely different from
taking a row times a column.
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So a column of A was -- what's
the shape of a column of A?
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m by one.
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A column of A is a column.
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It's got m entries
and one column.
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And what's a row of B?
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It's got one row and P columns.
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So what's the shape -- what do
I get if I multiply a column
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by a row?
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I get a big matrix.
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I get a full-sized matrix.
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If I multiply a
column by a row --
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should we just do one?
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Let me take the column two three
four times the row one six.
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That product there --
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I mean, when I'm just
following the rules of matrix
00:13:42.950 --> 00:13:46.490
multiplication, those rules
are just looking like --
00:13:46.490 --> 00:13:51.570
kind of petite, kind of
small, because the rows here
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are so short and the
columns there are so short,
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but they're the same
length, one entry.
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So what's the answer?
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What's the answer if I do two
three four times one six, just
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for practice?
00:14:05.320 --> 00:14:09.180
Well, what's the first
row of the answer?
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Two twelve.
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And the second row of the
answer is three eighteen.
00:14:16.420 --> 00:14:24.220
And the third row of the
answer is four twenty four.
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That's a very special
matrix, there.
00:14:29.400 --> 00:14:30.840
Very special matrix.
00:14:30.840 --> 00:14:32.830
What can you tell me
about its columns,
00:14:32.830 --> 00:14:34.220
the columns of that matrix?
00:14:37.640 --> 00:14:41.410
They're multiples
of this guy, right?
00:14:41.410 --> 00:14:42.690
They're multiples of that one.
00:14:42.690 --> 00:14:44.340
Which follows our rule.
00:14:44.340 --> 00:14:47.540
We said that the columns of
the answer were combinations,
00:14:47.540 --> 00:14:50.300
but there's only -- to take
a combination of one guy,
00:14:50.300 --> 00:14:52.300
it's just a multiple.
00:14:52.300 --> 00:14:54.000
The rows of the
answer, what can you
00:14:54.000 --> 00:14:55.530
tell me about those three rows?
00:14:58.060 --> 00:15:01.480
They're all multiples
of this row.
00:15:01.480 --> 00:15:04.660
They're all multiples of
one six, as we expected.
00:15:04.660 --> 00:15:06.970
But I'm getting a
full-sized matrix.
00:15:06.970 --> 00:15:15.800
And now, just to complete
this thought, if I have --
00:15:15.800 --> 00:15:17.280
let me write down
the fourth way.
00:15:21.110 --> 00:15:35.630
A B is a sum of columns
of A times rows of B.
00:15:35.630 --> 00:15:40.680
So that, for example, if my
matrix was two three four
00:15:40.680 --> 00:15:47.450
and then had another column,
say, seven eight nine,
00:15:47.450 --> 00:15:53.010
and my matrix here has -- say,
started with one six and then
00:15:53.010 --> 00:16:00.890
had another column
like zero zero, then --
00:16:00.890 --> 00:16:04.630
here's the fourth way, okay?
00:16:04.630 --> 00:16:07.690
I've got two columns there,
I've got two rows there.
00:16:07.690 --> 00:16:10.920
So the beautiful rule is --
00:16:10.920 --> 00:16:13.080
see, the whole thing
by columns and rows
00:16:13.080 --> 00:16:19.730
is that I can take the first
column times the first row
00:16:19.730 --> 00:16:25.100
and add the second column
times the second row.
00:16:32.690 --> 00:16:34.790
So that's the fourth way --
00:16:34.790 --> 00:16:38.760
that I can take
columns times rows,
00:16:38.760 --> 00:16:41.800
first column times first row,
second column times second
00:16:41.800 --> 00:16:42.960
row and add.
00:16:42.960 --> 00:16:44.110
Actually, what will I get?
00:16:44.110 --> 00:16:46.765
What will the answer be for
that matrix multiplication?
00:16:49.730 --> 00:16:52.130
Well, this one it's just
going to give us zero,
00:16:52.130 --> 00:16:56.070
so in fact I'm back to
this -- that's the answer,
00:16:56.070 --> 00:16:59.310
for that matrix multiplication.
00:16:59.310 --> 00:17:05.569
I'm happy to put up here
these facts about matrix
00:17:05.569 --> 00:17:10.040
multiplication, because it
gives me a chance to write down
00:17:10.040 --> 00:17:12.099
special matrices like this.
00:17:12.099 --> 00:17:15.050
This is a special matrix.
00:17:15.050 --> 00:17:18.569
All those rows lie
on the same line.
00:17:18.569 --> 00:17:21.900
All those rows lie on
the line through one six.
00:17:21.900 --> 00:17:25.240
If I draw a picture of
all these row vectors,
00:17:25.240 --> 00:17:28.119
they're all the same direction.
00:17:28.119 --> 00:17:30.730
If I draw a picture of
these two column vectors,
00:17:30.730 --> 00:17:33.900
they're in the same direction.
00:17:33.900 --> 00:17:37.430
Later, I would
use this language.
00:17:37.430 --> 00:17:39.310
Not too much later, either.
00:17:39.310 --> 00:17:42.400
I would say the
row space, which is
00:17:42.400 --> 00:17:44.380
like all the
combinations of the rows,
00:17:44.380 --> 00:17:47.060
is just a line for this matrix.
00:17:47.060 --> 00:17:51.240
The row space is the line
through the vector one six.
00:17:51.240 --> 00:17:54.390
All the rows lie on that line.
00:17:54.390 --> 00:17:57.710
And the column space
is also a line.
00:17:57.710 --> 00:18:01.510
All the columns lie on the
line through the vector two
00:18:01.510 --> 00:18:02.950
three four.
00:18:02.950 --> 00:18:07.340
So this is like a
really minimal matrix.
00:18:07.340 --> 00:18:10.940
And it's because of these ones.
00:18:10.940 --> 00:18:11.440
Okay.
00:18:11.440 --> 00:18:18.580
So that's a third way.
00:18:18.580 --> 00:18:23.160
Now I want to say one more thing
about matrix multiplication
00:18:23.160 --> 00:18:26.100
while we're on the subject.
00:18:26.100 --> 00:18:28.060
And it's this.
00:18:28.060 --> 00:18:29.740
You could also multiply --
00:18:29.740 --> 00:18:34.390
You could also cut
the matrix into blocks
00:18:34.390 --> 00:18:37.160
and do the
multiplication by blocks.
00:18:37.160 --> 00:18:46.690
Yet that's actually so, useful
that I want to mention it.
00:18:46.690 --> 00:18:47.745
Block multiplication.
00:18:50.540 --> 00:18:54.500
So I could take my matrix A
and I could chop it up, like,
00:18:54.500 --> 00:18:58.460
maybe just for simplicity,
let me chop it into two --
00:18:58.460 --> 00:18:59.770
into four square blocks.
00:18:59.770 --> 00:19:00.980
Suppose it's square.
00:19:00.980 --> 00:19:03.990
Let's just take a nice case.
00:19:03.990 --> 00:19:07.800
And B, suppose it's
square also, same size.
00:19:11.340 --> 00:19:13.460
So these sizes don't
have to be the same.
00:19:13.460 --> 00:19:16.540
What they have to do
is match properly.
00:19:16.540 --> 00:19:18.590
Here they certainly will match.
00:19:18.590 --> 00:19:22.340
So here's the rule for
block multiplication,
00:19:22.340 --> 00:19:28.880
that if this has
blocks like, A --
00:19:28.880 --> 00:19:34.220
so maybe A1, A2, A3,
A4 are the blocks here,
00:19:34.220 --> 00:19:38.210
and these blocks
are B1, B2,3 and B4?
00:19:38.210 --> 00:19:44.770
Then the answer
I can find block.
00:19:44.770 --> 00:19:46.500
And if you tell me
what's in that block,
00:19:46.500 --> 00:19:49.560
then I'm going to be quiet
about matrix multiplication
00:19:49.560 --> 00:19:51.870
for the rest of the day.
00:19:51.870 --> 00:19:54.920
What goes into that block?
00:19:54.920 --> 00:19:57.870
You see, these might be
-- this matrix might be --
00:19:57.870 --> 00:20:02.400
these matrices might be, like,
twenty by twenty with blocks
00:20:02.400 --> 00:20:06.030
that are ten by ten, to take the
easy case where all the blocks
00:20:06.030 --> 00:20:08.940
are the same shape.
00:20:08.940 --> 00:20:13.110
And the point is that I could
multiply those by blocks.
00:20:13.110 --> 00:20:15.940
And what goes in here?
00:20:15.940 --> 00:20:20.920
What's that block in the answer?
00:20:20.920 --> 00:20:25.710
A1 B1, that's a
matrix times a matrix,
00:20:25.710 --> 00:20:28.270
it's the right size, ten by ten.
00:20:28.270 --> 00:20:30.610
Any more?
00:20:30.610 --> 00:20:36.760
Plus, what else goes in there?
00:20:36.760 --> 00:20:38.160
A2 B3, right?
00:20:38.160 --> 00:20:41.490
It's just like block
rows times block columns.
00:20:45.610 --> 00:20:49.070
Nobody, I think, not even
Gauss could see instantly
00:20:49.070 --> 00:20:51.040
that it works.
00:20:51.040 --> 00:20:55.130
But somehow, if we check
it through, all five ways
00:20:55.130 --> 00:20:58.020
we're doing the same
multiplications.
00:20:58.020 --> 00:21:02.430
So this familiar
multiplication is
00:21:02.430 --> 00:21:04.270
what we're really
doing when we do it
00:21:04.270 --> 00:21:10.240
by columns, by rows by columns
times rows and by blocks.
00:21:10.240 --> 00:21:10.740
Okay.
00:21:10.740 --> 00:21:13.680
I just have to, like,
get the rules straight
00:21:13.680 --> 00:21:16.991
for matrix multiplication.
00:21:16.991 --> 00:21:17.490
Okay.
00:21:22.740 --> 00:21:24.490
All right, I'm ready
for the second topic,
00:21:24.490 --> 00:21:27.590
which is inverses.
00:21:27.590 --> 00:21:28.479
Okay.
00:21:28.479 --> 00:21:29.270
Ready for inverses.
00:21:34.140 --> 00:21:39.040
And let me do it for
square matrices first.
00:21:44.270 --> 00:21:44.770
Okay.
00:21:44.770 --> 00:21:52.270
So I've got a square matrix A.
00:21:52.270 --> 00:21:55.370
And it may or may not
have an inverse, right?
00:21:55.370 --> 00:21:57.260
Not all matrices have inverses.
00:21:57.260 --> 00:22:01.770
In fact, that's the most
important question you can ask
00:22:01.770 --> 00:22:06.260
about the matrix, is if it's
-- if you know it's square,
00:22:06.260 --> 00:22:08.540
is it invertible or not?
00:22:08.540 --> 00:22:12.500
If it is invertible, then
there is some other matrix,
00:22:12.500 --> 00:22:15.680
shall I call it A inverse?
00:22:15.680 --> 00:22:21.600
And what's the -- if
A inverse exists --
00:22:21.600 --> 00:22:24.270
there's a big "if" here.
00:22:24.270 --> 00:22:34.120
If this matrix exists, and it'll
be really central to figure out
00:22:34.120 --> 00:22:35.940
when does it exist?
00:22:35.940 --> 00:22:40.350
And then if it does exist,
how would you find it?
00:22:40.350 --> 00:22:45.710
But what's the equation
here that I haven't --
00:22:45.710 --> 00:22:47.900
that I have to finish now?
00:22:47.900 --> 00:22:53.280
This matrix, if it exists
multiplies A and produces,
00:22:53.280 --> 00:22:54.755
I think, the identity.
00:23:10.840 --> 00:23:12.680
But a real --
00:23:12.680 --> 00:23:17.600
an inverse for a square matrix
could be on the right as well
00:23:17.600 --> 00:23:19.240
--
00:23:19.240 --> 00:23:25.760
this is true, too, that it's --
00:23:25.760 --> 00:23:28.280
if I have a -- yeah in
fact, this is not --
00:23:28.280 --> 00:23:31.570
this is probably the --
00:23:31.570 --> 00:23:38.400
this is something that's not
easy to prove, but it works.
00:23:38.400 --> 00:23:40.240
That a left --
00:23:40.240 --> 00:23:43.670
square matrices, a left
inverse is also a right
00:23:43.670 --> 00:23:44.570
inverse.
00:23:44.570 --> 00:23:49.090
If I can find a matrix on the
left that gets the identity,
00:23:49.090 --> 00:23:50.990
then also that
matrix on the right
00:23:50.990 --> 00:23:53.630
will produce that identity.
00:23:53.630 --> 00:23:57.310
For rectangular matrices,
we'll see a left inverse
00:23:57.310 --> 00:23:58.950
that isn't a right inverse.
00:23:58.950 --> 00:24:01.660
In fact, the shapes
wouldn't allow it.
00:24:01.660 --> 00:24:03.700
But for square
matrices, the shapes
00:24:03.700 --> 00:24:09.820
allow it and it happens,
if A has an inverse.
00:24:09.820 --> 00:24:14.011
Okay, so give me some cases --
00:24:14.011 --> 00:24:14.510
let's see.
00:24:14.510 --> 00:24:17.090
I hate to be negative
here, but let's talk
00:24:17.090 --> 00:24:20.070
about the case with no inverse.
00:24:20.070 --> 00:24:31.210
So -- these matrices are called
invertible or non-singular --
00:24:36.640 --> 00:24:38.960
those are the good ones.
00:24:38.960 --> 00:24:41.510
And we want to be able
to identify how --
00:24:41.510 --> 00:24:44.180
if we're given a matrix,
has it got an inverse?
00:24:44.180 --> 00:24:46.950
Can I talk about
the singular case?
00:24:52.210 --> 00:24:52.940
No inverse.
00:24:57.420 --> 00:24:58.950
All right.
00:24:58.950 --> 00:25:02.720
Best to start with an example.
00:25:02.720 --> 00:25:06.880
Tell me an example -- let's
get an example up here.
00:25:06.880 --> 00:25:09.030
Let's make it two by two --
00:25:09.030 --> 00:25:13.540
of a matrix that has
not got an inverse.
00:25:13.540 --> 00:25:16.160
And let's see why.
00:25:16.160 --> 00:25:19.230
Let me write one up.
00:25:19.230 --> 00:25:20.950
No inverse.
00:25:20.950 --> 00:25:22.620
Let's see why.
00:25:22.620 --> 00:25:30.780
Let me write up --
one three two six.
00:25:35.180 --> 00:25:37.820
Why does that matrix
have no inverse?
00:25:40.950 --> 00:25:43.245
You could answer
that various ways.
00:25:45.760 --> 00:25:48.390
Give me one reason.
00:25:48.390 --> 00:25:51.620
Well, you could -- if you
know about determinants,
00:25:51.620 --> 00:25:57.060
which you're not supposed to,
you could take its determinant
00:25:57.060 --> 00:25:58.740
and you would get --
00:25:58.740 --> 00:26:00.330
Zero.
00:26:00.330 --> 00:26:03.090
Okay.
00:26:03.090 --> 00:26:04.120
Now -- all right.
00:26:07.640 --> 00:26:10.330
Let me ask you other reasons.
00:26:10.330 --> 00:26:13.120
I mean, as for other
reasons that that matrix
00:26:13.120 --> 00:26:15.810
isn't invertible.
00:26:15.810 --> 00:26:18.765
Here, I could use
what I'm saying here.
00:26:24.550 --> 00:26:27.800
Suppose A times other
matrix gave the identity.
00:26:32.060 --> 00:26:34.750
Why is that not possible?
00:26:34.750 --> 00:26:39.010
Because -- oh, yeah --
00:26:39.010 --> 00:26:41.220
I'm thinking about columns here.
00:26:41.220 --> 00:26:46.200
If I multiply this matrix A by
some other matrix, then the --
00:26:46.200 --> 00:26:50.240
the result -- what can you
tell me about the columns?
00:26:50.240 --> 00:26:55.730
They're all multiples
of those columns, right?
00:26:55.730 --> 00:26:59.340
If I multiply A by
another matrix that --
00:26:59.340 --> 00:27:04.130
the product has columns that
come from those columns.
00:27:04.130 --> 00:27:06.480
So can I get the
identity matrix?
00:27:06.480 --> 00:27:08.120
No way.
00:27:08.120 --> 00:27:11.670
The columns of the identity
matrix, like one zero --
00:27:11.670 --> 00:27:14.740
it's not a combination
of those columns,
00:27:14.740 --> 00:27:16.520
because those two
columns lie on the --
00:27:16.520 --> 00:27:19.140
both lie on the same line.
00:27:19.140 --> 00:27:22.140
Every combination is just
going to be on that line
00:27:22.140 --> 00:27:24.480
and I can't get one zero.
00:27:24.480 --> 00:27:33.530
So, do you see that sort of
column picture of the matrix
00:27:33.530 --> 00:27:34.750
not being invertible.
00:27:34.750 --> 00:27:37.580
In fact, here's another reason.
00:27:37.580 --> 00:27:40.720
This is even a more
important reason.
00:27:40.720 --> 00:27:42.260
Well, how can I
say more important?
00:27:42.260 --> 00:27:44.810
All those are important.
00:27:44.810 --> 00:27:47.910
This is another way to see it.
00:27:47.910 --> 00:27:51.410
A matrix has no inverse --
00:27:51.410 --> 00:27:55.710
yeah -- here -- now
this is important.
00:27:55.710 --> 00:27:59.180
A matrix has no -- a square
matrix won't have an inverse
00:27:59.180 --> 00:28:07.980
if there's no inverse
because I can solve --
00:28:07.980 --> 00:28:20.810
I can find an X of -- a
vector X with A times --
00:28:20.810 --> 00:28:23.140
this A times X giving zero.
00:28:27.010 --> 00:28:31.310
This is the reason I like best.
00:28:31.310 --> 00:28:33.170
That matrix won't
have an inverse.
00:28:33.170 --> 00:28:40.490
Can you -- well, let
me change I to U.
00:28:40.490 --> 00:28:46.780
So tell me a vector X that,
solves A X equals zero.
00:28:46.780 --> 00:28:49.610
I mean, this is, like,
the key equation.
00:28:49.610 --> 00:28:51.310
In mathematics, all
the key equations
00:28:51.310 --> 00:28:53.490
have zero on the
right-hand side.
00:28:53.490 --> 00:28:55.490
So what's the X?
00:28:55.490 --> 00:28:58.290
Tell me an X here --
00:28:58.290 --> 00:29:01.360
so now I'm going to put --
slip in the X that you tell me
00:29:01.360 --> 00:29:05.500
and I'm going to get zero.
00:29:05.500 --> 00:29:09.000
What X would do that job?
00:29:09.000 --> 00:29:11.370
Three and negative one?
00:29:11.370 --> 00:29:13.610
Is that the one you
picked, or -- yeah.
00:29:13.610 --> 00:29:18.720
Or another -- well, if
you picked zero with zero,
00:29:18.720 --> 00:29:21.060
I'm not so excited, right?
00:29:21.060 --> 00:29:23.490
Because that would always work.
00:29:23.490 --> 00:29:26.550
So it's really the
fact that this vector
00:29:26.550 --> 00:29:28.840
isn't zero that's important.
00:29:28.840 --> 00:29:33.750
It's a non-zero vector and
three negative one would do it.
00:29:33.750 --> 00:29:37.130
That just says three of this
column minus one of that column
00:29:37.130 --> 00:29:38.861
is the zero column.
00:29:38.861 --> 00:29:39.360
Okay.
00:29:42.240 --> 00:29:46.860
So now I know that A
couldn't be invertible.
00:29:46.860 --> 00:29:49.890
But what's the reasoning?
00:29:49.890 --> 00:29:54.720
If A X is zero, suppose I
multiplied by A inverse.
00:29:54.720 --> 00:29:56.660
Yeah, well here's the reason.
00:29:56.660 --> 00:30:02.180
Here -- this is why this
spells disaster for an inverse.
00:30:02.180 --> 00:30:06.580
The matrix can't have an
inverse if some combination
00:30:06.580 --> 00:30:09.140
of the columns gives
z- it gives nothing.
00:30:09.140 --> 00:30:12.630
Because, I could
take A X equals zero,
00:30:12.630 --> 00:30:21.200
I could multiply by A inverse
and what would I discover?
00:30:21.200 --> 00:30:23.840
Suppose I take that equation
and I multiply by --
00:30:23.840 --> 00:30:26.860
if A inverse existed, which
of course I'm going to come
00:30:26.860 --> 00:30:30.560
to the conclusion it can't
because if it existed,
00:30:30.560 --> 00:30:33.570
if there was an A inverse
to this dopey matrix,
00:30:33.570 --> 00:30:36.990
I would multiply that equation
by that inverse and I would
00:30:36.990 --> 00:30:42.270
discover X is zero.
00:30:42.270 --> 00:30:45.320
If I multiply A by A inverse
on the left, I get X.
00:30:45.320 --> 00:30:48.830
If I multiply by A inverse
on the right, I get zero.
00:30:48.830 --> 00:30:50.790
So I would discover X was zero.
00:30:50.790 --> 00:30:53.050
But it -- X is not zero.
00:30:53.050 --> 00:30:54.800
X -- this guy wasn't zero.
00:30:54.800 --> 00:30:55.460
There it is.
00:30:55.460 --> 00:30:58.170
It's three minus one.
00:30:58.170 --> 00:31:06.250
So, conclusion -- only, it takes
us some time to really work
00:31:06.250 --> 00:31:08.040
with that conclusion --
00:31:08.040 --> 00:31:14.300
our conclusion will be that
non-invertible matrices,
00:31:14.300 --> 00:31:19.490
singular matrices, some
combinations of their columns
00:31:19.490 --> 00:31:22.290
gives the zero column.
00:31:22.290 --> 00:31:26.090
They they take some
vector X into zero.
00:31:26.090 --> 00:31:30.650
And there's no way A
inverse can recover, right?
00:31:30.650 --> 00:31:32.520
That's what this equation says.
00:31:32.520 --> 00:31:36.980
This equation says I take
this vector X and multiplying
00:31:36.980 --> 00:31:39.470
by A gives zero.
00:31:39.470 --> 00:31:42.220
But then when I
multiply by A inverse,
00:31:42.220 --> 00:31:44.690
I can never escape from zero.
00:31:44.690 --> 00:31:47.980
So there couldn't
be an A inverse.
00:31:47.980 --> 00:31:51.701
Where here -- okay, now fix --
00:31:51.701 --> 00:31:52.200
all right.
00:31:52.200 --> 00:31:57.200
Now let me take -- all right,
back to the positive side.
00:31:57.200 --> 00:32:01.530
Let's take a matrix that
does have an inverse.
00:32:01.530 --> 00:32:03.840
And why not invert it?
00:32:03.840 --> 00:32:04.500
Okay.
00:32:04.500 --> 00:32:08.782
Can I -- so let me take on
this third board a matrix --
00:32:08.782 --> 00:32:09.990
shall I fix that up a little?
00:32:12.810 --> 00:32:15.660
Tell me a matrix that
has got an inverse.
00:32:18.680 --> 00:32:22.930
Well, let me say one three
two -- what shall I put there?
00:32:25.620 --> 00:32:28.665
Well, don't put six,
I guess is -- right?
00:32:31.570 --> 00:32:35.000
Do I any favorites here?
00:32:35.000 --> 00:32:36.910
One?
00:32:36.910 --> 00:32:38.280
Or eight?
00:32:41.470 --> 00:32:42.140
I don't care.
00:32:42.140 --> 00:32:43.450
What, seven?
00:32:43.450 --> 00:32:43.950
Seven.
00:32:43.950 --> 00:32:44.540
Okay.
00:32:44.540 --> 00:32:46.060
Seven is a lucky number.
00:32:46.060 --> 00:32:48.750
All right, seven, okay.
00:32:48.750 --> 00:32:49.380
Okay.
00:32:49.380 --> 00:32:51.080
So -- now what's our idea?
00:32:51.080 --> 00:32:53.740
We believe that this
matrix is invertible.
00:32:53.740 --> 00:32:57.060
Those who like determinants have
quickly taken its determinant
00:32:57.060 --> 00:32:59.320
and found it wasn't zero.
00:32:59.320 --> 00:33:04.610
Those who like columns,
and probably that --
00:33:04.610 --> 00:33:08.450
that department is not
totally popular yet --
00:33:08.450 --> 00:33:11.080
but those who like columns
will look at those two columns
00:33:11.080 --> 00:33:15.090
and say, hey, they point
in different directions.
00:33:15.090 --> 00:33:18.080
So I can get anything.
00:33:18.080 --> 00:33:19.860
Now, let me see, what do I mean?
00:33:19.860 --> 00:33:21.880
How I going to
computer A inverse?
00:33:21.880 --> 00:33:24.060
So A inverse --
00:33:24.060 --> 00:33:28.360
here's A inverse, now,
and I have to find it.
00:33:28.360 --> 00:33:33.219
And what do I get when I
do this multiplication?
00:33:33.219 --> 00:33:33.760
The identity.
00:33:40.810 --> 00:33:43.940
You know, forgive me for
taking two by two-s, but --
00:33:43.940 --> 00:33:49.240
lt's good to keep the
computations manageable and let
00:33:49.240 --> 00:33:51.090
the ideas come out.
00:33:51.090 --> 00:33:55.230
Okay, now what's
the idea I want?
00:33:55.230 --> 00:33:57.520
I'm looking for this
matrix A inverse, how
00:33:57.520 --> 00:33:58.750
I going to find it?
00:33:58.750 --> 00:34:04.235
Right now, I've got
four numbers to find.
00:34:08.290 --> 00:34:12.159
I'm going to look
at the first column.
00:34:12.159 --> 00:34:16.860
Let me take this
first column, A B.
00:34:16.860 --> 00:34:18.699
What's up there?
00:34:18.699 --> 00:34:20.670
What -- tell me this.
00:34:20.670 --> 00:34:25.260
What equation does the
first column satisfy?
00:34:25.260 --> 00:34:31.489
The first column satisfies A
times that column is one zero.
00:34:31.489 --> 00:34:34.179
The first column of the answer.
00:34:34.179 --> 00:34:38.820
And the second column,
C D, satisfies A times
00:34:38.820 --> 00:34:41.139
that second column is zero one.
00:34:41.139 --> 00:34:48.909
You see that finding the inverse
is like solving two systems.
00:34:48.909 --> 00:34:52.210
One system, when the
right-hand side is one zero --
00:34:52.210 --> 00:34:54.139
I'm just going to split
it into two pieces.
00:34:57.427 --> 00:34:58.760
I don't even need to rewrite it.
00:34:58.760 --> 00:35:04.480
I can take A times --
so let me put it here.
00:35:04.480 --> 00:35:18.100
A times column j of A inverse
is column j of the identity.
00:35:21.350 --> 00:35:23.010
I've got n equations.
00:35:23.010 --> 00:35:26.500
I've got, well,
two in this case.
00:35:26.500 --> 00:35:29.100
And they have the
same matrix, A,
00:35:29.100 --> 00:35:30.860
but they have different
right-hand sides.
00:35:30.860 --> 00:35:32.840
The right-hand sides
are just the columns
00:35:32.840 --> 00:35:35.800
of the identity, this
guy and this guy.
00:35:35.800 --> 00:35:37.450
And these are the two solutions.
00:35:37.450 --> 00:35:39.610
Do you see what I'm going --
00:35:39.610 --> 00:35:45.120
I'm looking at that
equation by columns.
00:35:45.120 --> 00:35:47.050
I'm looking at A
times this column,
00:35:47.050 --> 00:35:49.910
giving that guy, and A times
that column giving that guy.
00:35:49.910 --> 00:35:55.500
So -- Essentially -- so
this is like the Gauss --
00:35:55.500 --> 00:35:56.410
we're back to Gauss.
00:35:56.410 --> 00:36:00.820
We're back to solving systems of
equations, but we're solving --
00:36:00.820 --> 00:36:05.160
we've got two right-hand
sides instead of one.
00:36:05.160 --> 00:36:07.010
That's where Jordan comes in.
00:36:07.010 --> 00:36:09.980
So at the very beginning
of the lecture,
00:36:09.980 --> 00:36:13.230
I mentioned Gauss-Jordan,
let me write it up again.
00:36:13.230 --> 00:36:13.730
Okay.
00:36:13.730 --> 00:36:16.630
Here's the Gauss-Jordan idea.
00:36:21.600 --> 00:36:35.000
Gauss-Jordan solve
two equations at once.
00:36:39.460 --> 00:36:39.960
Okay.
00:36:39.960 --> 00:36:43.630
Let me show you how
the mechanics go.
00:36:43.630 --> 00:36:48.970
How do I solve a
single equation?
00:36:48.970 --> 00:36:55.760
So the two equations
are one three two seven,
00:36:55.760 --> 00:37:01.630
multiplying A B gives one zero.
00:37:01.630 --> 00:37:03.560
And the other
equation is the same
00:37:03.560 --> 00:37:12.130
one three two seven
multiplying C D gives zero one.
00:37:12.130 --> 00:37:15.010
Okay.
00:37:15.010 --> 00:37:17.410
That'll tell me the two
columns of the inverse.
00:37:17.410 --> 00:37:19.080
I'll have inverse.
00:37:19.080 --> 00:37:22.940
In other words, if I can
solve with this matrix A,
00:37:22.940 --> 00:37:24.850
if I can solve with
that right-hand side
00:37:24.850 --> 00:37:28.340
and that right-hand
side, I'm invertible.
00:37:28.340 --> 00:37:29.460
I've got it.
00:37:29.460 --> 00:37:30.690
Okay.
00:37:30.690 --> 00:37:36.410
And Jordan sort of said to
Gauss, solve them together,
00:37:36.410 --> 00:37:39.650
look at the matrix -- if
we just solve this one,
00:37:39.650 --> 00:37:43.140
I would look at one
three two seven,
00:37:43.140 --> 00:37:46.030
and how do I deal with
the right-hand side?
00:37:46.030 --> 00:37:48.960
I stick it on as an
extra column, right?
00:37:56.130 --> 00:37:58.460
That's this augmented matrix.
00:37:58.460 --> 00:38:01.750
That's the matrix when I'm
watching the right-hand side
00:38:01.750 --> 00:38:04.470
at the same time, doing the
same thing to the right side
00:38:04.470 --> 00:38:06.520
that I do to the left?
00:38:06.520 --> 00:38:09.610
So I just carry it along
as an extra column.
00:38:09.610 --> 00:38:12.100
Now I'm going to carry
along two extra columns.
00:38:17.200 --> 00:38:20.920
And I'm going to do
whatever Gauss wants, right?
00:38:20.920 --> 00:38:24.150
I'm going to do elimination.
00:38:24.150 --> 00:38:27.010
I'm going to get
this to be simple
00:38:27.010 --> 00:38:30.070
and this thing will
turn into the inverse.
00:38:30.070 --> 00:38:32.460
This is what's coming.
00:38:32.460 --> 00:38:35.800
I'm going to do elimination
steps to make this
00:38:35.800 --> 00:38:39.240
into the identity,
and lo and behold,
00:38:39.240 --> 00:38:41.550
the inverse will show up here.
00:38:41.550 --> 00:38:43.330
K--- let's do it.
00:38:43.330 --> 00:38:43.830
Okay.
00:38:43.830 --> 00:38:46.880
So what are the
elimination steps?
00:38:46.880 --> 00:38:51.680
So you see -- here's my matrix
A and here's the identity, like,
00:38:51.680 --> 00:38:52.795
stuck on, augmented on.
00:38:52.795 --> 00:38:53.670
STUDENT: I'm sorry...
00:38:53.670 --> 00:38:54.211
STRANG: Yeah?
00:38:54.211 --> 00:38:58.800
STUDENT: -- is the two and the
three supposed to be switched?
00:38:58.800 --> 00:39:01.070
STRANG: Did I -- oh, no,
they weren't supposed to be
00:39:01.070 --> 00:39:01.570
switched.
00:39:01.570 --> 00:39:02.070
Sorry.
00:39:02.070 --> 00:39:04.680
Thanks.
00:39:04.680 --> 00:39:05.930
Okay.
00:39:05.930 --> 00:39:08.425
Thank you very much.
00:39:08.425 --> 00:39:09.800
And there -- I've
got them right.
00:39:09.800 --> 00:39:11.215
Okay, thanks.
00:39:14.520 --> 00:39:15.510
Okay.
00:39:15.510 --> 00:39:17.780
So let's do elimination.
00:39:17.780 --> 00:39:19.890
All right, it's going
to be simple, right?
00:39:19.890 --> 00:39:23.690
So I take two of this
row away from this row.
00:39:23.690 --> 00:39:27.180
So this row stays
the same and two
00:39:27.180 --> 00:39:28.680
of those come away from this.
00:39:28.680 --> 00:39:32.560
That leaves me with a zero and
a one and two of these away from
00:39:32.560 --> 00:39:36.970
this is that what
you're getting --
00:39:36.970 --> 00:39:39.930
after one elimination step --
00:39:39.930 --> 00:39:42.790
Let me sort of separate the --
00:39:42.790 --> 00:39:45.040
the left half from
the right half.
00:39:45.040 --> 00:39:48.690
So two of that first row got
subtracted from the second row.
00:39:48.690 --> 00:39:53.240
Now this is an upper
triangular form.
00:39:53.240 --> 00:39:57.120
Gauss would quit, but
Jordan says keeps going.
00:39:57.120 --> 00:39:59.250
Use elimination upwards.
00:39:59.250 --> 00:40:03.710
Subtract a multiple of
equation two from equation one
00:40:03.710 --> 00:40:05.970
to get rid of the three.
00:40:05.970 --> 00:40:08.970
So let's go the whole way.
00:40:08.970 --> 00:40:14.420
So now I'm going to -- this guy
is fine, but I'm going to --
00:40:14.420 --> 00:40:15.770
what do I do now?
00:40:15.770 --> 00:40:19.850
What's my final step that
produces the inverse?
00:40:19.850 --> 00:40:22.200
I multiply this by
the right number
00:40:22.200 --> 00:40:26.060
to get up to ther to
remove that three.
00:40:26.060 --> 00:40:28.350
So I guess, I --
since this is a one,
00:40:28.350 --> 00:40:30.830
there's the pivot sitting there.
00:40:30.830 --> 00:40:33.750
I multiply it by three
and subtract from that,
00:40:33.750 --> 00:40:35.080
so what do I get?
00:40:35.080 --> 00:40:38.460
I'll have one zero -- oh,
yeah that was my whole point.
00:40:38.460 --> 00:40:41.640
I'll multiply this by three
and subtract from that,
00:40:41.640 --> 00:40:46.450
which will give me seven.
00:40:46.450 --> 00:40:49.340
And I multiply this by three
and subtract from that,
00:40:49.340 --> 00:40:50.920
which gives me a minus three.
00:41:00.060 --> 00:41:06.500
And what's my hope, belief?
00:41:06.500 --> 00:41:10.780
Here I started with
A and the identity,
00:41:10.780 --> 00:41:16.410
and I ended up with
the identity and who?
00:41:16.410 --> 00:41:18.640
That better be A inverse.
00:41:24.090 --> 00:41:26.980
That's the Gauss Jordan idea.
00:41:26.980 --> 00:41:33.690
Start with this long matrix,
double-length A I, eliminate,
00:41:33.690 --> 00:41:37.450
eliminate until this
part is down to I,
00:41:37.450 --> 00:41:40.530
then this one will --
must be for some reason,
00:41:40.530 --> 00:41:45.230
and we've got to find the
reason -- must be A inverse.
00:41:45.230 --> 00:41:46.690
Shall I just check
that it works?
00:41:50.160 --> 00:41:55.080
Let me just check that -- can I
multiply this matrix this part
00:41:55.080 --> 00:42:00.300
times A, I'll carry A
over here and just do that
00:42:00.300 --> 00:42:01.190
multiplication.
00:42:01.190 --> 00:42:03.310
You'll see I'll do it
the old fashioned way.
00:42:03.310 --> 00:42:06.000
Seven minus six is a one.
00:42:06.000 --> 00:42:08.670
Twenty one minus
twenty one is a zero,
00:42:08.670 --> 00:42:13.280
minus two plus two is a zero,
minus six plus seven is a one.
00:42:13.280 --> 00:42:15.780
Check.
00:42:15.780 --> 00:42:18.290
So that is the inverse.
00:42:18.290 --> 00:42:20.950
That's the Gauss-Jordan idea.
00:42:20.950 --> 00:42:24.190
So, you'll -- one of the
homework problems or more than
00:42:24.190 --> 00:42:30.710
one for Wednesday will ask
you to go through those steps.
00:42:30.710 --> 00:42:33.560
I think you just got to go
through Gauss-Jordan a couple
00:42:33.560 --> 00:42:38.600
of times, but I --
00:42:38.600 --> 00:42:43.690
yeah -- just to
see the mechanics.
00:42:43.690 --> 00:42:48.000
But the, important
thing is, why --
00:42:48.000 --> 00:42:50.190
is, like, what happened?
00:42:50.190 --> 00:42:53.570
Why did we -- why did
we get A inverse there?
00:42:53.570 --> 00:42:54.690
Let me ask you that.
00:42:57.860 --> 00:43:03.190
We got -- so we take --
00:43:03.190 --> 00:43:08.020
We do row reduction, we do
elimination on this long matrix
00:43:08.020 --> 00:43:12.840
A I until the first half
00:43:12.840 --> 00:43:14.990
Then a second half
is A inverse. is up.
00:43:14.990 --> 00:43:20.050
Well, how do I see that?
00:43:20.050 --> 00:43:22.310
Let me put up here
how I see that.
00:43:22.310 --> 00:43:30.100
So here's my Gauss-Jordan thing,
and I'm doing stuff to it.
00:43:30.100 --> 00:43:34.320
So I'm -- well,
whole lot of E's.
00:43:37.530 --> 00:43:40.150
Remember those are those
elimination matrices.
00:43:40.150 --> 00:43:42.861
Those are the -- those are the
things that we figured out last
00:43:42.861 --> 00:43:43.360
time.
00:43:43.360 --> 00:43:49.370
Yes, that's what an elimination
step is it's in matrix form,
00:43:49.370 --> 00:43:51.430
I'm multiplying by some Es.
00:43:51.430 --> 00:43:55.120
And the result -- well, so I'm
multiplying by a whole bunch
00:43:55.120 --> 00:43:55.650
of Es.
00:43:55.650 --> 00:43:57.530
So, I get a --
00:43:57.530 --> 00:44:02.070
can I call the overall matrix E?
00:44:02.070 --> 00:44:06.420
That's the elimination matrix,
the product of all those little
00:44:06.420 --> 00:44:06.920
pieces.
00:44:06.920 --> 00:44:09.050
What do I mean by little pieces?
00:44:09.050 --> 00:44:11.160
Well, there was an
elimination matrix
00:44:11.160 --> 00:44:14.400
that subtracted two of
that away from that.
00:44:14.400 --> 00:44:16.180
Then there was an
elimination matrix
00:44:16.180 --> 00:44:19.160
that subtracted three
of that away from that.
00:44:19.160 --> 00:44:21.370
I guess in this
case, that was all.
00:44:21.370 --> 00:44:24.280
So there were just two
Es in this case, one
00:44:24.280 --> 00:44:26.630
that did this step and
one that did this step
00:44:26.630 --> 00:44:31.690
and together they gave me
an E that does both steps.
00:44:31.690 --> 00:44:38.560
And the net result
was to get an I here.
00:44:38.560 --> 00:44:45.490
And you can tell me
what that has to be.
00:44:45.490 --> 00:44:50.240
This is, like, the
picture of what happened.
00:44:50.240 --> 00:44:53.350
If E multiplied A,
whatever that E is --
00:44:53.350 --> 00:44:59.780
we never figured
it out in this way.
00:44:59.780 --> 00:45:06.692
But whatever that E times
that E is, E times A is --
00:45:09.410 --> 00:45:12.120
What's E times A?
00:45:12.120 --> 00:45:13.480
It's I.
00:45:13.480 --> 00:45:20.510
That E, whatever the heck it
was, multiplied A and produced
00:45:20.510 --> 00:45:22.710
So E must be --
00:45:22.710 --> 00:45:30.640
E A equaling I tells us what
E is, I. namely it is --
00:45:30.640 --> 00:45:32.110
STUDENT: It's the inverse of A.
00:45:32.110 --> 00:45:33.720
STRANG: It's the inverse of A.
00:45:33.720 --> 00:45:35.660
Great.
00:45:35.660 --> 00:45:39.780
And therefore, when the second
half, when E multiplies I,
00:45:39.780 --> 00:45:41.470
it's E --
00:45:41.470 --> 00:45:44.760
Put this A inverse.
00:45:44.760 --> 00:45:48.450
You see the picture
looking that way?
00:45:48.450 --> 00:45:49.910
E times A is the identity.
00:45:49.910 --> 00:45:53.480
It tells us what E has to be.
00:45:53.480 --> 00:45:55.600
It has to be the
inverse, and therefore,
00:45:55.600 --> 00:45:57.870
on the right-hand
side, where E --
00:45:57.870 --> 00:46:00.580
where we just smartly
tucked on the identity,
00:46:00.580 --> 00:46:03.060
it's turning in, step by step --
00:46:03.060 --> 00:46:05.870
It's turning into A inverse.
00:46:05.870 --> 00:46:12.410
There is the statement of
Gauss-Jordan elimination.
00:46:12.410 --> 00:46:14.590
That's how you find the inverse.
00:46:14.590 --> 00:46:19.210
Where we can look at
it as elimination,
00:46:19.210 --> 00:46:24.260
as solving n equations
at the same time -- --
00:46:24.260 --> 00:46:28.170
and tacking on n columns,
solving those equations and up
00:46:28.170 --> 00:46:32.740
goes the n columns of
A inverse Okay, thanks.
00:46:32.740 --> 00:46:34.640
See you on Wednesday.