Okay, ready for me to start.

Ready for the taping to start in a minute.

He's going to raise his hand and signal when I'm on.

Just a minute, though, let them settle.0 Okay, guys.

Okay, give me the signal, then, when you want me to

start.

Okay, this is linear algebra, lecture four.

And, the first thing I have to do

is something that was on the list for last time,

but here it is now.

What's the inverse of a product?

If I multiply two matrices together

and I know their inverses, how do I

get the inverse of A times B?

So I know what inverses mean for a single matrix A

and for a matrix B.

What matrix do I multiply by to get the identity if I have A B?

Okay, that'll be simple but so basic.

Then I'm going to use that to --

I will have a product of matrices and the product

that we'll meet will be these elimination matrices

and the net result of today's lectures

is the big formula for elimination,

so the net result of today's lecture

is this great way to look at Gaussian elimination.

We know that we get from A to U by elimination.

We know the steps -- but now we get the right way to look

at it, A equals L U.

So that's the high point for today.

Okay.

Can I take the easy part, the first step first?

So, suppose A is invertible -- and of course it's going to be

a big question, when is the matrix invertible?

But let's say A is invertible and B is invertible,

then what matrix gives me the inverse of A B?

So that's the direct question.

What's the inverse of A B?

Do I multiply those separate inverses?

Yes.

I multiply the two matrices A inverse and B inverse,

but what order do I multiply?

In reverse order.

And you see why.

So the right thing to put here is B inverse A inverse.

That's the inverse I'm after.

We can just check that A B times that matrix gives the identity.

Okay.

So why -- once again, it's this fact that I can move

parentheses around.

I can just erase them all and do the multiplications any way

I want to.

So what's the right multiplication to do first?

B times B inverse.

This product here I is the identity.

Then A times the identity is the identity

and then finally A times A inverse gives the identity.

So forgive the dumb example in the book.

Why do you, do the inverse things in reverse order?

It's just like -- you take off your shoes,

you take off your socks, then the good way to invert that

process is socks back on first, then shoes.

Sorry, okay.

I'm sorry that's on the tape.

And, of course, on the other side we should really just

check -- on the other side I have B inverse,

A inverse.

That does multiply A B, and this time it's

these guys that give the identity,

squeeze down, they give the identity, we're in shape.

Okay.

So there's the inverse.

Good.

While we're at it, let me do a transpose,

because the next lecture has got a lot to --

involves transposes.

So how do I -- if I transpose a matrix,

I'm talking about square, invertible matrices right now.

If I transpose one, what's its inverse?

Well, the nice formula is --

let's see.

Let me start from A, A inverse equal the identity.

So -- give me a first row.

And let me transpose both sides.

That will bring a transpose into the picture.

So if I transpose the identity matrix, what do I have?

The identity, right?

If I exchange rows and columns, the identity

is a symmetric matrix.

It doesn't know the difference.

If I transpose these guys, that product, then again

it turns out that I have to reverse the order.

I can transpose them separately, but when I multiply,

those transposes come in the opposite order.

So it's A inverse transpose times A transpose

giving the identity.

So that's -- this equation is -- just comes directly from that

one.

But this equation tells me what I wanted to know,

namely what is the inverse of this guy A transpose?

What's the inverse of that -- if I transpose a matrix,

what'ss the inverse of the result?

And this equation tells me that here it is.

This is the inverse of A transpose.

Inverse of A transpose.

Of A transpose.

So I'll put a big circle around that,

because that's the answer, that's the best

answer we could hope for.

That if you want to know the inverse of A transpose

and you know the inverse of A, then you just transpose that.

So in a -- to put it another way,

transposing and inversing you can do in either order

for a single matrix.

Okay.

So these are like basic facts that we can now use,

all right -- so now I put it to use.

I put it to use by thinking -- we're really completing,

the subject of elimination.

Actually, -- the thing about elimination is it's the right

way to understand what the matrix has got.

This A equal L U is the most basic factorization

of a matrix.

I always worry that you will think

this course is all elimination.

It's just row operations.

And please don't.

We'll be beyond that, but it's the right algebra to do first.

Okay.

So, now I'm coming near the end of it,

but I want to get it in a decent form.

So my decent form is matrix form.

I have a matrix A, let's suppose it's a good matrix,

I can do elimination, no row exchanges --

So no row exchanges for now.

Pivots all fine, nothing zero in the pivot position.

I get to the very end, which is U.

So I get from A to U.

And I want to know what's the connection?

STUDENT: Zero one zero?

How is A related to U?

And this is going to tell me that there's

a matrix L that connects them.

Okay.

Can I do it for a two by two first?

Okay.

Two by two, elimination.

Okay, so I'll do it under here.

Okay.

So let my matrix A be --

We'll keep it simple, say two and an eight,

so we know that the first pivot is a two,

and the multiplier's going to be a four

and then let me put a one here and what number

do I not want to put there?

Four.

I don't want a four there, because in that case,

the second pivot would not -- we wouldn't have a second pivot.

The matrix would be singular, general screw-up.

Okay.

So let me put some other number here like seven.

Okay.

Okay.

Now I want to operate on that with my elementary matrix.

So what's the elementary matrix?

Strictly speaking, it's E21, because it's

the guy that's going to produce a zero in that position.

And it's going to produce U in one shot,

because it's just a two by two matrix.

So two one and I'm going to take four of those away from those,

produce that zero and leave a three there.

And that's U.

And what's the matrix that did it?

Quick review, then.

What's the elimination elementary matrix E21 --

it's one zero, thanks.

And -- negative four one,

right.

Good.

Okay.

So that -- you see the difference between this

and what I'm shooting for.

I'm shooting for A on one side and the other matrices

on the other side of the equation.

Okay.

So I can do that right away.

Now here's going to be my A equals L U.

And you won't have any trouble telling me what --

so A is still two one eight seven.

L is what you're going to tell me and U is still two one

zero three.

Okay.

So what's L in this case?

Well, first -- so how is L related to this E guy?

It's the inverse, because I want to multiply through

by the inverse of this, which will put the identity here,

and the inverse will show up there and I'll call it L.

So what is the inverse of this?

Remember those elimination matrices are easy to invert.

The inverse matrix for this one is 1 0 4 1, it's actually flip

STRANG: Zero one zero. sign.

Okay.

Do you want --

if we did the numbers right, we must --

this should be correct.

Okay.

And of course it is.

That says the first row's right, four times the first row

plus the second row is eight seven.

Good.

That's simple, two by two.

Okay.

But it already shows the form that we're headed for.

It shows -- so what's the L stand for?

Why the letter L?

If U stood for upper triangular, then of course L

stands for lower triangular.

And actually, it has ones on the diagonal, where this thing has

the pivots on the diagonal.

Oh, sometimes we may want to separate out the pivots,

so can I just mention that sometimes we could also write

this as --

we could have this one zero four one --

I'll just show you how I would divide out this matrix

of pivots -- two three.

There's a diagonal matrix.

And I just -- whatever is left is here.

Now what's left?

If I divide this first row by two to pull out the two,

then I have a one and a one half.

And if I divide the second row by three to pull out the three,

then I have a one.

Where shall we put them?4 So if this is L U,

this is maybe called L D or pivot U.

And now it's a little more balanced,

because we have ones on the diagonal here and here.

And the diagonal matrix in the middle.

So both of those...

Matlab would produce either one.

I'll basically stay with L U.

Okay.

Now I have to think about bigger than two by two.

But right now, this was just like easy exercise.

And, to tell the truth, this one was a minus sign

and this one was a plus sign.

I mean, that's the only difference.

But, with three by three, there's

a more significant difference.

Let me show you how that works.

Let me move up to a three by three,

let's say some matrix A, okay?

Let's imagine it's three by three.

I won't write numbers down for now.

So what's the first elimination step

that I do, the first matrix I multiply it by,

what letter will I use for

that?

It'll be E two one, because it's --

the first step will be to get a zero in that two one position.

right?

Okay, now a second row -- say zero zero one and the third guy