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.

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?

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 has the plus sign because it adds back

what this removes.

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. Okay.

That's simple, two by two.

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.

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?

And then the next step will be to get a zero in the three one position.

And the final step will be to get a zero in the three two

That's what elimination is, and it produced U. position.

And again, no row exchanges.

I'm taking the nice case, now, the typical case, too -- when I don't have to do any row

exchange, all I do is these elimination steps.

Okay.

Now, suppose I want that stuff over on the right-hand side, as I really do.

That's, like, my point here.

I can multiply these together to get a matrix E, but I want it over on the right.

I want its inverse over there.

So what's the right expression now?

If I write A and U, what goes there?

Okay.

So I've got the inverse of this, I've got three matrices in

a row now.

And it's their inverses that are going to show up, because each one is easy to invert.

Question is, what about the whole bunch?

How easy is it to invert the whole bunch?

So, that's what we know how to do.

We know how to invert, we should take the separate inverses, but they go in the opposite

order.

So what goes here?

E three two inverse, right, because I'll multiply from the left by E three two inverse, then

I'll pop it up next to U.

And then will come E three one inverse.

And then this'll be the only guy left standing and that's gone when I do an E two one inverse.

So there is L.

That's L U.

L is product of inverses.

Now you still can ask why is this guy preferring inverses?

And let me explain why.

Let me explain why is this product nicer than this one?

This product turns out to be better than this one.

Let me take a typical case here.

Let me take a typical case.

So let me -- I have to do three by three for you to see the improvement.

Two by two, it was just one E, no problem.

But let me go up to this case.

Suppose my matrices E21 -- suppose E21 has a minus two in there.

Suppose that -- and now suppose -- oh, I'll even suppose E31 is the identity.

I'm going to make the point with just a couple of these.

Okay.

Now this guy will have -- do something -- now let's suppose minus five one.

Okay. There's typical.

That's a typical case in which we didn't need an E31. Maybe we already had a zero in that