Here's the idea: If

A is a n by

n matrix, and it

has an inverse, I will say

a bit more about that later, then

the inverse is going to

be written A to the

minus one and A

times this inverse, A to

the minus one, is going to

equal to A inverse times

A, is going to

give us back the identity matrix.

Okay?

Only matrices that are

m by m for some the idea of M having inverse.

So, a matrix is

M by M, this is also

called a square matrix and

it's called square because

the number of rows is equal to the number of columns.

Right and it turns out

only square matrices have inverses,

so A is a square

matrix, is m by m,

on inverse this equation over here.

Let's look at a concrete example,

so let's say I

have a matrix, three, four,

two, sixteen.

So this is a two by

two matrix, so it's

a square matrix and so this

may just could have an and

it turns out that I

happen to know the inverse

of this matrix is zero point

four, minus zero point

one, minus zero point

zero five, zero zero seven five.

And if I take this matrix

and multiply these together it

turns out what I get

is the two by

two identity matrix, I,

this is I two by two.

Okay?

And so on this slide,

you know this matrix is

the matrix A, and this matrix is the matrix A-inverse.

And it turns out

if that you are computing A

times A-inverse, it turns out

if you compute A-inverse times

A you also get back the identity matrix.

So how did I

find this inverse or how

did I come up with this inverse over here?

It turns out that sometimes

you can compute inverses by hand

but almost no one does that these days.

And it turns out there is

very good numerical software for

taking a matrix and computing its inverse.

So again, this is one of

those things where there are lots

of open source libraries that

you can link to from any

of the popular programming languages to compute inverses of matrices.

Let me show you a quick example.

How I actually computed this inverse,

and what I did was I used software called Optive.

So let me bring that up.

We will see a lot about Optive later.

Let me just quickly show you an example.

Set my matrix A to

be equal to that matrix on

the left, type three four

two sixteen, so that's my matrix A right.

This is matrix 34,

216 that I have down

here on the left.

And, the software lets me compute

the inverse of A very easily.

It's like P over A equals this.

And so, this is right,

this matrix here on my

four minus, on my one, and so on.

This given the numerical

solution to what is the

inverse of A. So let me

just write, inverse of A

equals P inverse of

A over that I

can now just verify that A

times A inverse the identity

is, type A times the

inverse of A and

the result of that is

this matrix and this is

one one on the diagonal

and essentially ten to

the minus seventeen, ten to the

minus sixteen, so Up to

numerical precision, up to

a little bit of round off

error that my computer

had in finding optimal matrices

and these numbers off the

diagonals are essentially zero

so A times the inverse is essentially the identity matrix.

Can also verify the inverse of

A times A is also

equal to the identity,

ones on the diagonals and values

that are essentially zero except

for a little bit of round

dot error on the off diagonals.