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I've been multiplying matrices
already, but certainly time for
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me to discuss the rules 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
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multiplying a matrix B and --
giving me a result -- well,
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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 -- I might -- maybe
I take it C 3 4,
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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
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four, as you know.
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Column four.
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And can I just write down,
or can we write down the
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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 -- I could use
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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, use a little matrix
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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 -- most of the
course, I use whole vectors.
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I very seldom,
get down to the details of
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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,
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on along -- 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 -- when are
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we allowed to multiply these
matrices?
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What are the shapes of these
things?
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The shapes are -- if we allow
them to be not necessarily
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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.
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 that we meet
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coming down has to match the
number of ones across.
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So B will have to be n by
something.
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Whatever.
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P.
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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.
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Okay.
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So there are m times P little
numbers in there,
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entries, 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 to look at that same
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calculation, 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.
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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 -- here's the second way to
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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 that part of the answer.
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The matrix times the second
column is the second column of
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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 by a
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vector, which I already knew how
to do, and I can think of just P
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columns sitting side by side,
just like resting next to each
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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, to be able to think,
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okay, matrix multiplication
works so that I can just think
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of having several columns,
multiplying by A and getting
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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, and A times a vector
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is a combination of the columns
of A.
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And it makes sense,
because the columns of A have
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length m and the columns of C
have length m.
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And every column of C is some
combination of the columns of A.
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And it's these numbers in here
that tell me what combination it
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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 -- a row of A multiplying all
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these rows here and producing a
row of the product.
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So this row takes a combination
of these rows and that's the
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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,
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it's got its rows and I
multiply by A,
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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 -- so that's --
now we've got,
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like, the regular way,
the column way,
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the row way and -- what's left?
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The one that I can -- well,
one way is columns times rows.
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What happens if I multiply --
So this was row times column,
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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, that's definitely different
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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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n 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
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column 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
-- should we just do one?
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Let me take the column two
three four times the row one
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six.
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That product there --
I mean, when I'm just following
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the rules of matrix
multiplication,
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those rules are just looking
like -- kind of petite,
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kind of small,
because the rows here are so
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short and the columns there are
so short, but they're the same
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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,
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just for practice?
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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.
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And the third row of the answer
is four twenty four.
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That's a very special matrix,
there.
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Very special matrix.
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What can you tell me about its
columns, the columns of that
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matrix?
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They're multiples of this guy,
right?
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They're multiples of that one.
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Which follows our rule.
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We said that the columns of the
answer were combinations,
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but there's only -- to take a
combination of one guy,
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it's just a multiple.
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The rows of the answer,
what can you tell me about
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those three rows?
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They're all multiples of this
row.
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They're all multiples of one
six, as we expected.
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But I'm getting a full-sized
matrix.
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And now, just to complete this
thought, if I have -- let me
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write down the fourth way.
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A B is a sum of columns of A
times rows of B.
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So that, for example,
if my matrix was two three four
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and then had another column,
say, seven eight nine,
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and my matrix here has -- say,
started with one six and then
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had another column like zero
zero, then -- here's the fourth
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way, okay?
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I've got two columns there,
I've got two rows there.
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So the beautiful rule is --
see, the whole thing by columns
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and rows is that I can take the
first column times the first row
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and add the second column times
the second row.
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So that's the fourth way --
that I can take columns times
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rows, first column times first
row, second column times second
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row and add.
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Actually, what will I get?
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What will the answer be for
that matrix multiplication?
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Well, this one it's just going
to give us zero,
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so in fact I'm back to this --
that's the answer,
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for that matrix multiplication.
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I'm happy to put up here these
facts about matrix
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multiplication,
because it gives me a chance to
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write down special matrices like
this.
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This is a special matrix.
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All those rows lie on the same
line.
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All those rows lie on the line
through one six.
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If I draw a picture of all
these row vectors,
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they're all the same direction.
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If I draw a picture of these
two column vectors,
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they're in the same direction.
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Later, I would use this
language.
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Not too much later,
either.
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I would say the row space,
which is like all the
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combinations of the rows,
is just a line for this matrix.
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The row space is the line
through the vector one six.
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All the rows lie on that line.
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And the column space is also a
line.
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All the columns lie on the line
through the vector two three
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four.
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So this is like a really
minimal matrix.
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And it's because of these ones.
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Okay.
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So that's a third way.
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Now I want to say one more
thing about matrix
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multiplication while we're on
the subject.
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And it's this.
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You could also multiply --
You could also cut the matrix
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into blocks and do the
multiplication by blocks.
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Yet that's actually so,
useful that I want to mention
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it.
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Block multiplication.
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So I could take my matrix A and
I could chop it up,
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like, maybe just for
simplicity, let me chop it into
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two --
into four square blocks.
283
00:18:59 --> 00:19:01.17
Suppose it's square.
284
00:19:01.17 --> 00:19:03.94
Let's just take a nice case.
285
00:19:03.94 --> 00:19:08
And B, suppose it's square
also, same size.
286
00:19:08 --> 00:19:12
So these sizes don't have to be
the same.
287
00:19:12 --> 00:19:16
What they have to do is match
properly.
288
00:19:16 --> 00:19:20.23
Here they certainly will match.
289
00:19:20.23 --> 00:19:25
So here's the rule for block
multiplication,
290
00:19:25 --> 00:19:30
that if this has blocks like,
A -- so maybe A1,
291
00:19:30 --> 00:19:36
A2, A3, A4 are the blocks here,
and these blocks are B1,
292
00:19:36 --> 00:19:37.98
B2,3 and B4?
293
00:19:37.98 --> 00:19:42
Then the answer I can find
block.
294
00:19:42 --> 00:19:46
And if you tell me what's in
that block, then I'm going to be
295
00:19:46 --> 00:19:48
quiet about matrix
multiplication for the rest of
296
00:19:48 --> 00:19:49
the day.
297
00:19:49 --> 00:19:50
What goes into that block?
298
00:19:50 --> 00:19:53
You see, these might be -- this
matrix might be -- these
299
00:19:53 --> 00:19:55
matrices might be,
like, twenty by twenty with
300
00:19:55 --> 00:19:59.17
blocks that are ten by ten,
to take the easy case where all
301
00:19:59.17 --> 00:20:01
the blocks are the same shape.
302
00:20:01 --> 00:20:08
And the point is that I could
multiply those by blocks.
303
00:20:08 --> 00:20:11.47
And what goes in here?
304
00:20:11.47 --> 00:20:15
What's that block in the
answer?
305
00:20:15 --> 00:20:23.12
A1 B1, that's a matrix times a
matrix, it's the right size,
306
00:20:23.12 --> 00:20:25
ten by ten.
307
00:20:25 --> 00:20:26
Any more?
308
00:20:26 --> 00:20:30
Plus, what else goes in there?
309
00:20:30 --> 00:20:32
A2 B3, right?
310
00:20:32 --> 00:20:37
It's just like block rows times
block columns.
311
00:20:37 --> 00:20:42
Nobody, I think,
not even Gauss could see
312
00:20:42 --> 00:20:46
instantly that it works.
313
00:20:46 --> 00:20:50.29
But somehow,
if we check it through,
314
00:20:50.29 --> 00:20:55
all five ways we're doing the
same multiplications.
315
00:20:55 --> 00:21:01
So this familiar multiplication
is what we're really doing when
316
00:21:01 --> 00:21:06.33
we do it by columns,
by rows by columns times rows
317
00:21:06.33 --> 00:21:08
and by blocks.
318
00:21:08 --> 00:21:09
Okay.
319
00:21:09 --> 00:21:15
I just have to,
like, get the rules straight
320
00:21:15 --> 00:21:19
for matrix multiplication.
321
00:21:19 --> 00:21:20.24
Okay.
322
00:21:20.24 --> 00:21:26
All right, I'm ready for the
second topic,
323
00:21:26 --> 00:21:28
which is inverses.
324
00:21:28 --> 00:21:29
Okay.
325
00:21:29 --> 00:21:32
Ready for inverses.
326
00:21:32 --> 00:21:40
And let me do it for square
matrices first.
327
00:21:40 --> 00:21:40
Okay.
328
00:21:40 --> 00:21:43
So I've got a square matrix A.
329
00:21:43 --> 00:21:47
And it may or may not have an
inverse, right?
330
00:21:47 --> 00:21:50.52
Not all matrices have inverses.
331
00:21:50.52 --> 00:21:55
In fact, that's the most
important question you can ask
332
00:21:55 --> 00:21:59
about the matrix,
is if it's -- if you know it's
333
00:21:59 --> 00:22:03
square, is it invertible or not?
334
00:22:03 --> 00:22:10
If it is invertible,
then there is some other
335
00:22:10 --> 00:22:14
matrix, shall I call it A
inverse?
336
00:22:14 --> 00:22:23
And what's the -- if A inverse
exists -- there's a big "if"
337
00:22:23 --> 00:22:25
here.
338
00:22:25 --> 00:22:31
If this matrix exists,
and it'll be really central to
339
00:22:31 --> 00:22:34
figure out when does it exist?
340
00:22:34 --> 00:22:40.3
And then if it does exist,
how would you find it?
341
00:22:40.3 --> 00:22:47
But what's the equation here
that I haven't -- that I have to
342
00:22:47 --> 00:22:49
finish now?
343
00:22:49 --> 00:23:00
This matrix,
if it exists multiplies A and
344
00:23:00 --> 00:23:10
produces, I think,
the identity.
345
00:23:10 --> 00:23:17
But a real -- an inverse for a
square matrix could be on the
346
00:23:17 --> 00:23:24
right as well -- this is true,
too, that it's -- if I have a
347
00:23:24 --> 00:23:30.28
-- yeah in fact,
this is not -- this is probably
348
00:23:30.28 --> 00:23:34
the --
this is something that's not
349
00:23:34 --> 00:23:36
easy to prove,
but it works.
350
00:23:36 --> 00:23:42
That a left -- square matrices,
a left inverse is also a right
351
00:23:42 --> 00:23:42
inverse.
352
00:23:42 --> 00:23:47
If I can find a matrix on the
left that gets the identity,
353
00:23:47 --> 00:23:52
then also that matrix on the
right will produce that
354
00:23:52 --> 00:23:54
identity.
355
00:23:54 --> 00:23:58
For rectangular matrices,
we'll see a left inverse that
356
00:23:58 --> 00:24:00
isn't a right inverse.
357
00:24:00 --> 00:24:03
In fact, the shapes wouldn't
allow it.
358
00:24:03 --> 00:24:08
But for square matrices,
the shapes allow it and it
359
00:24:08 --> 00:24:10.68
happens, if A has an inverse.
360
00:24:10.68 --> 00:24:15
Okay, so give me some cases --
let's see.
361
00:24:15 --> 00:24:21
I hate to be negative here,
but let's talk about the case
362
00:24:21 --> 00:24:23
with no inverse.
363
00:24:23 --> 00:24:30
So -- these matrices are called
invertible or non-singular --
364
00:24:30 --> 00:24:33.27
those are the good ones.
365
00:24:33.27 --> 00:24:39
And we want to be able to
identify how -- if we're given a
366
00:24:39 --> 00:24:44
matrix, has it got an inverse?
367
00:24:44 --> 00:24:49
Can I talk about the singular
case?
368
00:24:49 --> 00:24:50
No inverse.
369
00:24:50 --> 00:24:51
All right.
370
00:24:51 --> 00:24:56
Best to start with an example.
371
00:24:56 --> 00:25:02
Tell me an example -- let's get
an example up here.
372
00:25:02 --> 00:25:10
Let's make it two by two -- of
a matrix that has not got an
373
00:25:10 --> 00:25:11
inverse.
374
00:25:11 --> 00:25:14
And let's see why.
375
00:25:14 --> 00:25:18
Let me write one up.
376
00:25:18 --> 00:25:20.69
No inverse.
377
00:25:20.69 --> 00:25:23
Let's see why.
378
00:25:23 --> 00:25:29
Let me write up -- one three
two six.
379
00:25:29 --> 00:25:36
Why does that matrix have no
inverse?
380
00:25:36 --> 00:25:42
You could answer that various
ways.
381
00:25:42 --> 00:25:45
Give me one reason.
382
00:25:45 --> 00:25:54.11
Well, you could --
if you know about determinants,
383
00:25:54.11 --> 00:26:00
which you're not supposed to,
you could take its determinant
384
00:26:00 --> 00:26:03
and you would get -- Zero.
385
00:26:03 --> 00:26:03
Okay.
386
00:26:03 --> 00:26:05
Now -- all right.
387
00:26:05 --> 00:26:08
Let me ask you other reasons.
388
00:26:08 --> 00:26:14
I mean, as for other reasons
that that matrix isn't
389
00:26:14 --> 00:26:16
invertible.
390
00:26:16 --> 00:26:21
Here, I could use what I'm
saying here.
391
00:26:21 --> 00:26:27
Suppose A times other matrix
gave the identity.
392
00:26:27 --> 00:26:30
Why is that not possible?
393
00:26:30 --> 00:26:35
Because -- oh,
yeah -- I'm thinking about
394
00:26:35 --> 00:26:37
columns here.
395
00:26:37 --> 00:26:43
If I multiply this matrix A by
some other matrix,
396
00:26:43 --> 00:26:48
then the --
the result -- what can you tell
397
00:26:48 --> 00:26:50
me about the columns?
398
00:26:50 --> 00:26:54
They're all multiples of those
columns, right?
399
00:26:54 --> 00:26:59
If I multiply A by another
matrix that -- the product has
400
00:26:59 --> 00:27:02.38
columns that come from those
columns.
401
00:27:02.38 --> 00:27:05
So can I get the identity
matrix?
402
00:27:05 --> 00:27:06
No way.
403
00:27:06 --> 00:27:10
The columns of the identity
matrix, like one zero -- it's
404
00:27:10 --> 00:27:14
not a combination of those
columns, because those two
405
00:27:14 --> 00:27:18
columns lie on the -- both lie
on the same line.
406
00:27:18 --> 00:27:22
Every combination is just going
to be on that line and I can't
407
00:27:22 --> 00:27:24
get one zero.
408
00:27:24 --> 00:27:30
So, do you see that sort of
column picture of the matrix not
409
00:27:30 --> 00:27:32
being invertible.
410
00:27:32 --> 00:27:35
In fact, here's another reason.
411
00:27:35 --> 00:27:38
This is even a more important
reason.
412
00:27:38 --> 00:27:42
Well, how can I say more
important?
413
00:27:42 --> 00:27:44
All those are important.
414
00:27:44 --> 00:27:48
This is another way to see it.
415
00:27:48 --> 00:27:55
A matrix has no inverse -- yeah
-- here -- now this is
416
00:27:55 --> 00:27:56
important.
417
00:27:56 --> 00:28:04
A matrix has no -- a square
matrix won't have an inverse if
418
00:28:04 --> 00:28:12
there's no inverse because I can
solve -- I can find an X of -- a
419
00:28:12 --> 00:28:20
vector X with A times -- this A
times X giving zero.
420
00:28:20 --> 00:28:24
This is the reason I like best.
421
00:28:24 --> 00:28:29
That matrix won't have an
inverse.
422
00:28:29 --> 00:28:35
Can you -- well,
let me change I to U.
423
00:28:35 --> 00:28:42
So tell me a vector X that,
solves A X equals zero.
424
00:28:42 --> 00:28:49.88
I mean, this is,
like, the key equation.
425
00:28:49.88 --> 00:28:53
In mathematics,
all the key equations have zero
426
00:28:53 --> 00:28:55
on the right-hand side.
427
00:28:55 --> 00:28:56
So what's the X?
428
00:28:56 --> 00:29:01
Tell me an X here -- so now I'm
going to put -- slip in the X
429
00:29:01 --> 00:29:04
that you tell me and I'm going
to get zero.
430
00:29:04 --> 00:29:07
What X would do that job?
431
00:29:07 --> 00:29:09
Three and negative one?
432
00:29:09 --> 00:29:12
Is that the one you picked,
or -- yeah.
433
00:29:12 --> 00:29:16
Or another -- well,
if you picked zero with zero,
434
00:29:16 --> 00:29:18.2
I'm not so excited,
right?
435
00:29:18.2 --> 00:29:20
Because that would always work.
436
00:29:20 --> 00:29:25
So it's really the fact that
this vector isn't zero that's
437
00:29:25 --> 00:29:27
important.
438
00:29:27 --> 00:29:32
It's a non-zero vector and
three negative one would do it.
439
00:29:32 --> 00:29:37
That just says three of this
column minus one of that column
440
00:29:37 --> 00:29:39
is the zero column.
441
00:29:39 --> 00:29:39
Okay.
442
00:29:39 --> 00:29:43
So now I know that A couldn't
be invertible.
443
00:29:43 --> 00:29:45
But what's the reasoning?
444
00:29:45 --> 00:29:49
If A X is zero,
suppose I multiplied by A
445
00:29:49 --> 00:29:50
inverse.
446
00:29:50 --> 00:29:53
Yeah, well here's the reason.
447
00:29:53 --> 00:29:59
Here -- this is why this spells
disaster for an inverse.
448
00:29:59 --> 00:30:04
The matrix can't have an
inverse if some combination of
449
00:30:04 --> 00:30:09
the columns gives z- it gives
nothing.
450
00:30:09 --> 00:30:12
Because, I could take A X
equals zero, I could multiply by
451
00:30:12 --> 00:30:14
A inverse and what would I
discover?
452
00:30:14 --> 00:30:18
Suppose I take that equation
and I multiply by -- if A
453
00:30:18 --> 00:30:20
inverse existed,
which of course I'm going to
454
00:30:20 --> 00:30:24
come to the conclusion it can't
because if it existed,
455
00:30:24 --> 00:30:27
if there was an A inverse to
this dopey matrix,
456
00:30:27 --> 00:30:30
I would multiply that equation
by that inverse and I would
457
00:30:30 --> 00:30:32
discover X is zero.
458
00:30:32 --> 00:30:38
If I multiply A by A inverse on
the left, I get X.
459
00:30:38 --> 00:30:45
If I multiply by A inverse on
the right, I get zero.
460
00:30:45 --> 00:30:49
So I would discover X was zero.
461
00:30:49 --> 00:30:52
But it -- X is not zero.
462
00:30:52 --> 00:30:54
X -- this guy wasn't zero.
463
00:30:54 --> 00:30:55
There it is.
464
00:30:55 --> 00:30:57
It's three minus one.
465
00:30:57 --> 00:31:01
So, conclusion -- only,
it takes us some time to really
466
00:31:01 --> 00:31:05
work with that conclusion -- our
conclusion will be that
467
00:31:05 --> 00:31:08
non-invertible matrices,
singular matrices,
468
00:31:08 --> 00:31:13
some combinations of their
columns gives the zero column.
469
00:31:13 --> 00:31:17
They they take some vector X
into zero.
470
00:31:17 --> 00:31:23
And there's no way A inverse
can recover, right?
471
00:31:23 --> 00:31:26
That's what this equation says.
472
00:31:26 --> 00:31:33
This equation says I take this
vector X and multiplying by A
473
00:31:33 --> 00:31:35
gives zero.
474
00:31:35 --> 00:31:41
But then when I multiply by A
inverse, I can never escape from
475
00:31:41 --> 00:31:42
zero.
476
00:31:42 --> 00:31:45
So there couldn't be an A
inverse.
477
00:31:45 --> 00:31:49
Where here -- okay,
now fix -- all right.
478
00:31:49 --> 00:31:56
Now let me take -- all right,
back to the positive side.
479
00:31:56 --> 00:32:00.35
Let's take a matrix that does
have an inverse.
480
00:32:00.35 --> 00:32:02
And why not invert it?
481
00:32:02 --> 00:32:02
Okay.
482
00:32:02 --> 00:32:08
Can I -- so let me take on this
third board a matrix -- shall I
483
00:32:08 --> 00:32:10
fix that up a little?
484
00:32:10 --> 00:32:14
Tell me a matrix that has got
an inverse.
485
00:32:14 --> 00:32:20
Well, let me say one three two
-- what shall I put there?
486
00:32:20 --> 00:32:27
Well, don't put six,
I guess is -- right?
487
00:32:27 --> 00:32:31
Do I any favorites here?
488
00:32:31 --> 00:32:31
One?
489
00:32:31 --> 00:32:33
Or eight?
490
00:32:33 --> 00:32:35
I don't care.
491
00:32:35 --> 00:32:37
What, seven?
492
00:32:37 --> 00:32:38.61
Seven.
493
00:32:38.61 --> 00:32:39
Okay.
494
00:32:39 --> 00:32:45
Seven is a lucky number.
495
00:32:45 --> 00:32:46
All right, seven,
okay.
496
00:32:46 --> 00:32:47
Okay.
497
00:32:47 --> 00:32:48.84
So -- now what's our idea?
498
00:32:48.84 --> 00:32:51
We believe that this matrix is
invertible.
499
00:32:51 --> 00:32:55
Those who like determinants
have quickly taken its
500
00:32:55 --> 00:32:57
determinant and found it wasn't
zero.
501
00:32:57 --> 00:33:01
Those who like columns,
and probably that --
502
00:33:01 --> 00:33:06
that department is not totally
popular yet -- but those who
503
00:33:06 --> 00:33:11
like columns will look at those
two columns and say,
504
00:33:11 --> 00:33:14
hey, they point in different
directions.
505
00:33:14 --> 00:33:16
So I can get anything.
506
00:33:16 --> 00:33:20
Now, let me see,
what do I mean?
507
00:33:20 --> 00:33:24
How I going to computer A
inverse?
508
00:33:24 --> 00:33:28
So A inverse -- here's A
inverse, now,
509
00:33:28 --> 00:33:30
and I have to find it.
510
00:33:30 --> 00:33:35
And what do I get when I do
this multiplication?
511
00:33:35 --> 00:33:37.4
The identity.
512
00:33:37.4 --> 00:33:42.31
You know, forgive me for taking
two by two-s,
513
00:33:42.31 --> 00:33:46
but --
lt's good to keep the
514
00:33:46 --> 00:33:52
computations manageable and let
the ideas come out.
515
00:33:52 --> 00:33:55
Okay, now what's the idea I
want?
516
00:33:55 --> 00:34:02
I'm looking for this matrix A
inverse, how I going to find it?
517
00:34:02 --> 00:34:07
Right now, I've got four
numbers to find.
518
00:34:07 --> 00:34:12
I'm going to look at the first
column.
519
00:34:12 --> 00:34:15
Let me take this first column,
A B.
520
00:34:15 --> 00:34:17
What's up there?
521
00:34:17 --> 00:34:19
What -- tell me this.
522
00:34:19 --> 00:34:24
What equation does the first
column satisfy?
523
00:34:24 --> 00:34:32
The first column satisfies A
times that column is one zero.
524
00:34:32 --> 00:34:34
The first column of the answer.
525
00:34:34 --> 00:34:38.97
And the second column,
C D, satisfies A times that
526
00:34:38.97 --> 00:34:41.15
second column is zero one.
527
00:34:41.15 --> 00:34:45
You see that finding the
inverse is like solving two
528
00:34:45 --> 00:34:46
systems.
529
00:34:46 --> 00:34:51
One system, when the right-hand
side is one zero -- I'm just
530
00:34:51 --> 00:34:54
going to split it into two
pieces.
531
00:34:54 --> 00:34:59
I don't even need to rewrite
it.
532
00:34:59 --> 00:35:07
I can take A times -- so let me
put it here.
533
00:35:07 --> 00:35:16
A times column j of A inverse
is column j of the identity.
534
00:35:16 --> 00:35:20.2
I've got n equations.
535
00:35:20.2 --> 00:35:27.07
I've got, well,
two in this case.
536
00:35:27.07 --> 00:35:30
And they have the same matrix,
A, but they have different
537
00:35:30 --> 00:35:32
right-hand sides.
538
00:35:32 --> 00:35:36
The right-hand sides are just
the columns of the identity,
539
00:35:36 --> 00:35:37
this guy and this guy.
540
00:35:37 --> 00:35:39
And these are the two
solutions.
541
00:35:39 --> 00:35:43
Do you see what I'm going --
I'm looking at that equation by
542
00:35:43 --> 00:35:45
columns.
543
00:35:45 --> 00:35:48
I'm looking at A times this
column, giving that guy,
544
00:35:48 --> 00:35:50
and A times that column giving
that guy.
545
00:35:50 --> 00:35:54
So -- Essentially -- so this is
like the Gauss -- we're back to
546
00:35:54 --> 00:35:55
Gauss.
547
00:35:55 --> 00:35:58
We're back to solving systems
of equations,
548
00:35:58 --> 00:36:02
but we're solving -- we've got
two right-hand sides instead of
549
00:36:02 --> 00:36:03
one.
550
00:36:03 --> 00:36:08
That's where Jordan comes in.
551
00:36:08 --> 00:36:17
So at the very beginning of the
lecture, I mentioned
552
00:36:17 --> 00:36:24
Gauss-Jordan,
let me write it up again.
553
00:36:24 --> 00:36:25
Okay.
554
00:36:25 --> 00:36:30
Here's the Gauss-Jordan idea.
555
00:36:30 --> 00:36:39
Gauss-Jordan solve two
equations at once.
556
00:36:39 --> 00:36:39
Okay.
557
00:36:39 --> 00:36:44
Let me show you how the
mechanics go.
558
00:36:44 --> 00:36:48
How do I solve a single
equation?
559
00:36:48 --> 00:36:54
So the two equations are one
three two seven,
560
00:36:54 --> 00:36:58.98
multiplying A B gives one zero.
561
00:36:58.98 --> 00:37:06
And the other equation is the
same one three two seven
562
00:37:06 --> 00:37:11
multiplying C D gives zero one.
563
00:37:11 --> 00:37:12.21
Okay.
564
00:37:12.21 --> 00:37:16.66
That'll tell me the two columns
of the inverse.
565
00:37:16.66 --> 00:37:18
I'll have inverse.
566
00:37:18 --> 00:37:22.85
In other words,
if I can solve with this matrix
567
00:37:22.85 --> 00:37:27
A, if I can solve with that
right-hand side and that
568
00:37:27 --> 00:37:30
right-hand side,
I'm invertible.
569
00:37:30 --> 00:37:31
I've got it.
570
00:37:31 --> 00:37:33
Okay.
571
00:37:33 --> 00:37:38
And Jordan sort of said to
Gauss, solve them together,
572
00:37:38 --> 00:37:43
look at the matrix -- if we
just solve this one,
573
00:37:43 --> 00:37:49
I would look at one three two
seven, and how do I deal with
574
00:37:49 --> 00:37:51
the right-hand side?
575
00:37:51 --> 00:37:54
I stick it on as an extra
column, right?
576
00:37:54 --> 00:37:58
That's this augmented matrix.
577
00:37:58 --> 00:38:03
That's the matrix when I'm
watching the right-hand side at
578
00:38:03 --> 00:38:06
the same time,
doing the same thing to the
579
00:38:06 --> 00:38:09
right side that I do to the
left?
580
00:38:09 --> 00:38:12
So I just carry it along as an
extra column.
581
00:38:12 --> 00:38:16.71
Now I'm going to carry along
two extra columns.
582
00:38:16.71 --> 00:38:21
And I'm going to do whatever
Gauss wants, right?
583
00:38:21 --> 00:38:23
I'm going to do elimination.
584
00:38:23 --> 00:38:28
I'm going to get this to be
simple and this thing will turn
585
00:38:28 --> 00:38:30
into the inverse.
586
00:38:30 --> 00:38:32
This is what's coming.
587
00:38:32 --> 00:38:36.66
I'm going to do elimination
steps to make this into the
588
00:38:36.66 --> 00:38:41
identity, and lo and behold,
the inverse will show up here.
589
00:38:41 --> 00:38:43
K--- let's do it.
590
00:38:43 --> 00:38:44
Okay.
591
00:38:44 --> 00:38:46
So what are the elimination
steps?
592
00:38:46 --> 00:38:50
So you see -- here's my matrix
A and here's the identity,
593
00:38:50 --> 00:38:52
like, stuck on,
augmented on.
594
00:38:52 --> 00:38:53
STUDENT: I'm sorry...
595
00:38:53 --> 00:38:54
STRANG: Yeah?
596
00:38:54 --> 00:38:59
STUDENT: -- is the two and the
three supposed to be switched?
597
00:38:59 --> 00:39:03
STRANG: Did I -- oh,
no, they weren't supposed to be
598
00:39:03 --> 00:39:04
switched.
599
00:39:04 --> 00:39:05
Sorry.
600
00:39:05 --> 00:39:05
Thanks.
601
00:39:05 --> 00:39:06
Okay.
602
00:39:06 --> 00:39:08
Thank you very much.
603
00:39:08 --> 00:39:11
And there -- I've got them
right.
604
00:39:11 --> 00:39:12
Okay, thanks.
605
00:39:12 --> 00:39:12
Okay.
606
00:39:12 --> 00:39:15
So let's do elimination.
607
00:39:15 --> 00:39:20
All right, it's going to be
simple, right?
608
00:39:20 --> 00:39:23
So I take two of this row away
from this row.
609
00:39:23 --> 00:39:27
So this row stays the same and
two of those come away from
610
00:39:27 --> 00:39:28
this.
611
00:39:28 --> 00:39:33
That leaves me with a zero and
a one and two of these away from
612
00:39:33 --> 00:39:38
this is that what you're getting
-- after one elimination step --
613
00:39:38 --> 00:39:43
Let me sort of separate the --
the left half from the right
614
00:39:43 --> 00:39:44
half.
615
00:39:44 --> 00:39:49
So two of that first row got
subtracted from the second row.
616
00:39:49 --> 00:39:52.69
Now this is an upper triangular
form.
617
00:39:52.69 --> 00:39:56
Gauss would quit,
but Jordan says keeps going.
618
00:39:56 --> 00:39:59
Use elimination upwards.
619
00:39:59 --> 00:40:04
Subtract a multiple of equation
two from equation one to get rid
620
00:40:04 --> 00:40:05.24
of the three.
621
00:40:05.24 --> 00:40:07
So let's go the whole way.
622
00:40:07 --> 00:40:11
So now I'm going to -- this guy
is fine, but I'm going to --
623
00:40:11 --> 00:40:12
what do I do now?
624
00:40:12 --> 00:40:16
What's my final step that
produces the inverse?
625
00:40:16 --> 00:40:20
I multiply this by the right
number to get up to ther to
626
00:40:20 --> 00:40:22
remove that three.
627
00:40:22 --> 00:40:26
So I guess, I -- since this is
a one, there's the pivot sitting
628
00:40:26 --> 00:40:27
there.
629
00:40:27 --> 00:40:30
I multiply it by three and
subtract from that,
630
00:40:30 --> 00:40:31.46
so what do I get?
631
00:40:31.46 --> 00:40:35
I'll have one zero -- oh,
yeah that was my whole point.
632
00:40:35 --> 00:40:38
I'll multiply this by three and
subtract from that,
633
00:40:38 --> 00:40:41
which will give me seven.
634
00:40:41 --> 00:40:50
And I multiply this by three
and subtract from that,
635
00:40:50 --> 00:40:56
which gives me a minus three.
636
00:40:56 --> 00:41:00
And what's my hope,
belief?
637
00:41:00 --> 00:41:11
Here I started with A and the
identity, and I ended up with
638
00:41:11 --> 00:41:16
the identity and who?
639
00:41:16 --> 00:41:19
That better be A inverse.
640
00:41:19 --> 00:41:22
That's the Gauss Jordan idea.
641
00:41:22 --> 00:41:28
Start with this long matrix,
double-length A I,
642
00:41:28 --> 00:41:33
eliminate, eliminate until this
part is down to I,
643
00:41:33 --> 00:41:38
then this one will -- must be
for some reason,
644
00:41:38 --> 00:41:44
and we've got to find the
reason -- must be A inverse.
645
00:41:44 --> 00:41:49
Shall I just check that it
works?
646
00:41:49 --> 00:41:54
Let me just check that -- can I
multiply this matrix this part
647
00:41:54 --> 00:41:58
times A, I'll carry A over here
and just do that multiplication.
648
00:41:58 --> 00:42:01
You'll see I'll do it the old
fashioned way.
649
00:42:01 --> 00:42:03
Seven minus six is a one.
650
00:42:03 --> 00:42:07
Twenty one minus twenty one is
a zero, minus two plus two is a
651
00:42:07 --> 00:42:11
zero, minus six plus seven is a
one.
652
00:42:11 --> 00:42:11
Check.
653
00:42:11 --> 00:42:14
So that is the inverse.
654
00:42:14 --> 00:42:17
That's the Gauss-Jordan idea.
655
00:42:17 --> 00:42:22
So, you'll -- one of the
homework problems or more than
656
00:42:22 --> 00:42:28
one for Wednesday will ask you
to go through those steps.
657
00:42:28 --> 00:42:34
I think you just got to go
through Gauss-Jordan a couple of
658
00:42:34 --> 00:42:40
times, but I --
yeah -- just to see the
659
00:42:40 --> 00:42:41
mechanics.
660
00:42:41 --> 00:42:47.45
But the, important thing is,
why -- is, like,
661
00:42:47.45 --> 00:42:49
what happened?
662
00:42:49 --> 00:42:55
Why did we -- why did we get A
inverse there?
663
00:42:55 --> 00:42:57.66
Let me ask you that.
664
00:42:57.66 --> 00:43:04
We got -- so we take --
We do row reduction,
665
00:43:04 --> 00:43:11.19
we do elimination on this long
matrix A I until the first half
666
00:43:11.19 --> 00:43:11
is up.
667
00:43:11 --> 00:43:15
Then a second half is A
inverse.
668
00:43:15 --> 00:43:18
Well, how do I see that?
669
00:43:18 --> 00:43:22
Let me put up here how I see
that.
670
00:43:22 --> 00:43:28.26
So here's my Gauss-Jordan
thing, and I'm doing stuff to
671
00:43:28.26 --> 00:43:29
it.
672
00:43:29 --> 00:43:32
So I'm -- well,
whole lot of E's.
673
00:43:32 --> 00:43:36
Remember those are those
elimination matrices.
674
00:43:36 --> 00:43:41
Those are the -- those are the
things that we figured out last
675
00:43:41 --> 00:43:42
time.
676
00:43:42 --> 00:43:47
Yes, that's what an elimination
step is it's in matrix form,
677
00:43:47 --> 00:43:50
I'm multiplying by some Es.
678
00:43:50 --> 00:43:54
And the result -- well,
so I'm multiplying by a whole
679
00:43:54 --> 00:43:55
bunch of Es.
680
00:43:55 --> 00:43:59
So, I get a -- can I call the
overall matrix E?
681
00:43:59 --> 00:44:05
That's the elimination matrix,
the product of all those little
682
00:44:05 --> 00:44:05
pieces.
683
00:44:05 --> 00:44:09
What do I mean by little
pieces?
684
00:44:09 --> 00:44:14
Well, there was an elimination
matrix that subtracted two of
685
00:44:14 --> 00:44:15
that away from that.
686
00:44:15 --> 00:44:20.74
Then there was an elimination
matrix that subtracted three of
687
00:44:20.74 --> 00:44:22
that away from that.
688
00:44:22 --> 00:44:25
I guess in this case,
that was all.
689
00:44:25 --> 00:44:30
So there were just two Es in
this case, one that did this
690
00:44:30 --> 00:44:36
step and one that did this step
and together they gave me an E
691
00:44:36 --> 00:44:38
that does both steps.
692
00:44:38 --> 00:44:41
And the net result was to get
an I here.
693
00:44:41 --> 00:44:45
And you can tell me what that
has to be.
694
00:44:45 --> 00:44:50
This is, like,
the picture of what happened.
695
00:44:50 --> 00:44:57
If E multiplied A,
whatever that E is -- we never
696
00:44:57 --> 00:45:01
figured it out in this way.
697
00:45:01 --> 00:45:10
But whatever that E times that
E is, E times A is -- What's E
698
00:45:10 --> 00:45:11
times A?
699
00:45:11 --> 00:45:14
It's I.
700
00:45:14 --> 00:45:20.25
That E, whatever the heck it
was, multiplied A and produced
701
00:45:20.25 --> 00:45:20
I.
702
00:45:20 --> 00:45:25
So E must be -- E A equaling I
tells us what E is,
703
00:45:25 --> 00:45:30.67
namely it is -- STUDENT:
It's the inverse of A.
704
00:45:30.67 --> 00:45:33
STRANG: It's the inverse of A.
705
00:45:33 --> 00:45:34
Great.
706
00:45:34 --> 00:45:38
And therefore,
when the second half,
707
00:45:38 --> 00:45:42
when E multiplies I,
it's E --
708
00:45:42 --> 00:45:44.09
Put this A inverse.
709
00:45:44.09 --> 00:45:46
You see the picture looking
that way?
710
00:45:46 --> 00:45:48
E times A is the identity.
711
00:45:48 --> 00:45:50.22
It tells us what E has to be.
712
00:45:50.22 --> 00:45:52
It has to be the inverse,
and therefore,
713
00:45:52 --> 00:45:56
on the right-hand side,
where E -- where we just
714
00:45:56 --> 00:45:59
smartly tucked on the identity,
it's turning in,
715
00:45:59 --> 00:46:04
step by step --
It's turning into A inverse.
716
00:46:04 --> 00:46:09
There is the statement of
Gauss-Jordan elimination.
717
00:46:09 --> 00:46:13
That's how you find the
inverse.
718
00:46:13 --> 00:46:19
Where we can look at it as
elimination, as solving n
719
00:46:19 --> 00:46:26
equations at the same time --
-- and tacking on n columns,
720
00:46:26 --> 00:46:33
solving those equations and up
goes the n columns of A inverse
721
00:46:33 --> .
722
. --> 00:46:33.73
723
00:46:33.73 --> 00:46:35
Okay, thanks.
724
00:46:35 --> 00:46:38
See you on Wednesday.