I've been multiplying matrices already, but certainly time for me

to discuss the rules for matrix multiplication.

And the interesting part is the many ways you can do it,

and they all give the same answer. So it's -- and they're

all important. So matrix multiplication,

and then, come inverses. So we're --

we -- mentioned the inverse of a matrix, but there's --

that's a big deal. Lots to do about inverses

and how to find them. Okay, so I'll begin with how to

multiply two matrices. First way,

okay, so suppose I have a matrix A multiplying a matrix B and --

giving me a result -- well, I could call it C.

A times B. Okay. So, l- let me just review the

rule for w- for this entry. That's the entry in

row I and column J. So that's the I J entry.

Right there is C I J. We always write the row number and

then the column number. So I might --

I might -- maybe I take it C three four, just to make it specific.

So instead of I J, let me use numbers. C three four.

So where does that come from, the three four entry?

It comes from row three, here, row three and column four,

as you know. Column four.

And can I just write down, or can we write down the formula for

it? C three four is --

if we look at the whole row and the whole

column, the quick way for me to say it is row three of A --

I could use a dot for dot product. I won't often use that,

actually. S- dot column four of B.

And -- but this gives us a chance to just, like, use a little

matrix notation. What are the entries?

What's this first entry in row three?

The first -- the first -- that number that's

sitting right there is A,

so it's got two indices and what are they? [Class response,

inaudible] Three one. So there's an A three one there.

Now what's the first guy at the top of column four? [Class

response, inaudible] So what's sitting up there?

[Class response, inaudible] B one four, right.

So that this dot product starts with A three one times B one four.

And then what's the next -- so this is like I-

I'm accumulating this sum, then comes the next guy, A three two,

second column, times B two four, second row.

So it's B A three two, B two four and so on.

Just practice with indices. Oh, let -- let me even practice with

a summation formula. So this is --

I -- I -- most of the course, I use whole vectors.

I very seldom, get down to the details of these

particular entries, but here we'd better do it.

So I'm -- it's some kind of a sright?

Of things in row three, column K shall I say?

Times things in row K, column four. Do you see that that's what's --

that's what we're seeing here? This is K is one,

here K is two, on along -- so up --

so the sum goes all the way along the row and down the column,

say, one to N. So that's what the A --

the C three four entry looks like. A sum of A three K B K four.

Just takes a little practice to do that.

Okay. And, uh -- oh, well,

maybe I should say -- when are we allowed to multiply

these matrices? What are the shapes of these things?

The shapes are, uh -- if we allow them to be not

necessarily square matrices. If they're square,

they've got to be the same size. If they're rectangular, they're

not the same size. If they're rectangular,

this might be -- well, I always think of A as M by N.

M rows, N columns. So that sum goes to N.

Now what's the point -- how many rows does B have to have?

[Class response, inaudible] N. N.

The number of rows in B, the number of guys that we meet

coming down has to match the number of ones across.

So B will have to be N by something. Whatever. P.

So then -- the number of columns here has to match the number of rows

there, and then what's the result? What's the shape of the result?

What's the shape of C, the output? Well, it's got these same M rows,

or -- it's got M rows. And how many columns?

[Class response, inaudible] P. M by P. Okay.

So there are M times P little numbers in there,

entries, and each one, looks like that.

Okay. So that's the standard rule. That's the way people think of

multiplying matrices. I do it too.

But that's -- that's -- I want to talk about other

ways to look at that same calculation,

looking at whole columns and whole rows.

Okay. So can I do A B C again? A B equaling C again?

But now, tell me s- tell me about, yeah.

Let me -- I'll put it up here. So here goes A, again,

times B producing C. And again,

this is M by N. This is N by P and this is M by P.

Okay. Now I want to look

at whole columns. I want to look at the columns of --

here -- in fact - here's the second way to multiply matrices.

Because I'm going to build on what I know already.

How do I multiply a matrix by a column?

How do I -- I know how to multiply this matrix

by that column. Th- th- shall I call

that column one? That tells me column

one of the answer. The matrix times the first column is

that first column. Because none of this stuff entered

that part of the answer. The matrix times the second column

is the second column of the answer. Do you see what I'm saying?

That I could think of multiplying a matrix by a vector,

which I already knew how to do, and I can think of

the vec- I can think of just P columns sitting side by side,

just like resting next to each other.

And I multiply A times each one of those.

And I get the P columns of the answer.

Do you see this as -- this is quite nice, to be able to

think, okay, matrix multiplication works so

that I can just think of having several columns,

multiplying by A and getting the columns of the answer.

So, like, here's column one -- a -- shall I call that --

here's a -- shall I call that column one?

And what's going in there is A times column one.

Okay. So that's the picture a column at a time.

So what does that tell me? What does that tell me

about these columns? These columns of C are combinations,

because we've seen that before, of columns of A.

Every one of these comes from A times this,

and A times a vector is a combination of the columns of A.

And -- right -- and it makes sense,

because the columns of A have length M and the columns of

C have length M. And every column of C is a --

is some combination of the columns of A.

And it's these numbers in here that tell me what combination it is.

Do you see that? That out --

that in that answer, C, I'm seeing stuff that's col-

that's combinations of these columns.

Now, suppose I look at it -- that's two ways now.

The third way is look at it by rows. So now let me change to rows. Okay.

So now I can think of a row of A -- a row of A multiplying all these

rows here and producing a row of the product.

So this row takes a combination of these rows and that's the answer.

So these rows of C are combinations of what?

Of -- tell me how to finish that. The rows of C,

when I have a matrix B, it's got it's rows and I multiply by

A, and what does that do? It mixes the rows up.

It makes -- it creates combinations of the rows

of -- [student response, inaudible] -- B, thanks.

Rows of B. That's what I wanted to see,

that this -- that this answer -- I can see where the pieces are

coming from. The rows in the answer are coming as

combinations of these rows. The columns in the answer are com-

coming as combinations of those columns.

And now that's -- so that's three ways.

Now you can say, okay, what's the fourth way?

The fourth way -- so that's -- now we've got,

like, the regular way, the column way, the row way and -- what's left?

The -- the one that I can -- I -- I want to tell you about --

well, one way is columns times rows. What happens if I multiply --

so th- this was row times column, it gave a number.

Okay. Now I want to ask you about column times row.

What does -- if I multiply a column of A times a

row of B, what shape am I ending up with?

So if I take a column times a row, that's definitely different from

taking a row times a column. So a column of A was --

what -- what's the shape of a column of A?

N by one. A column of A is a column.

It's got M entries and one column. And what's a row of B?

It's got one row and P columns. So what's the shape --

what do I get if I multiply a column by a row?

I get a big matrix. I get a full-sized matrix.

If I multiply a column by a row, I get -- should we just do one?

Let me take the column two three four times the row one six.

That is a -- that product there -- I mean, when I'm just following the

rules of matrix multiplication,

those rules are just looking like -- kind of

petite, kind of small, because the -- the rows here are so

short and the columns there are so short, but they're the same

length, one entry. So what's the answer?

What's the answer if I do two three four times one six,

just for practice? Well, what's the first row of the

answer? Two twelve. And the second row of the answer is

three eighteen. And the third row of the answer is

four twenty four. Actually, what am I --

I mean, that's a very special matrix, there.

Very special matrix. What can you tell me about its

columns, the columns of that matrix?

They're multiples of this guy, right?

They're multiples of that one. Which follows our rule.

We said that the columns of the answer were combinations,

but there's only -- to take a combination of one guy,

it's just a multiple. The rows of the answer,

what can you tell me about those three rows?

They're all multiples of this row. They're all multiples of

one six, as we expected. But I'm getting a full-sized

matrix. And now, just to complete this

thought, if I have,-- now, l- let me right

down the fourth way. A B is a sum of columns

of A times rows of B. So that, for example, if my --

if my matrix was two three four and then had another column,

say, seven eight nine, and my matrix here has --

say, started with one six and then had another column like zero zero,

then -- h- here's the fourth way, okay?

I've got two columns there, I've got two rows there.

So the beautiful rule is -- see, the whole thing by columns

and rows is that I can take the first column times the first

row and add the second column times the second row.

So that's the fourth way, that -- that I can take columns times rows,

first column times first row, second column times second row and add.

Actually, what will I get? What will the answer be for that

matrix multiplication? Well, this one it's just going to

give us zero, so in fact I'm back to this --

that's the answer, for that matrix multiplication. [sneeze]

Uh -- I'm -- I'm sort of,

like, happy to put up here these facts about matrix multiplication,

because it gives me a chance to write down special

matrices like this. This is a special matrix.

All those rows lie on the same line.

All those rows lie on the line through one six.

If I draw a picture of all these row vectors, they're all

the same direction. If I draw a picture of these two

column vectors, they're in the same direction.

Later, I would use this language. Not too much later, either.

I would say the row space, which is like all the combinations

of the rows, is just a line for this matrix.

The row space is the line through the vector one six.

All the rows lie on that line. And the column space is also a line.

All the columns lie on the line through the vector two three four.

So this is like a really minimal matrix.

And it's because of these ones. Okay. So that's a third way.

Now, even -- yeI -- can I --

will you -- would you take -- this is -- I- I want to say one more

thing about matrix multiplication while we're on the subject.