- [Instructor] We know that when we're just multiplying
regular numbers we have the notion of a reciprocal.
For example, if I were to take two and I were to multiply it
by its reciprocal, it would be equal to one.
Or if I were to just take a
and a is not equal to zero and I were to multiply it
by its reciprocal for any a, that is not equal to zero
this will also be equal to one.
And this is a number that if I multiply times anything
I am just going to get that original number.
So that's interesting, put in the back of our minds
you learned this many, many years ago.
Now we also have something
that comes out of our knowledge of functions.
We know that if there's some function
let's call it f(x) that goes from some set,
we call that our domain
to some other set we call that our range,
that in many cases, not all cases
so this is the function f that goes from x to f(x).
That in many cases, but not always the case
there's another function that can take us back.
And we call that other function, the inverse of f.
So that if you apply the inverse of f to f(x)
you're going to get back to where you were.
You're going to get back to x.
And we also know that it goes the other way around.
For example, if you did f of f inverse of x,
that too will get us back to x.
So the natural question is
is there an analog for an inverse of a function,
or for reciprocal when we're multiplying
when we think about matrices.
So let's play with a few ideas.
So let's imagine a matrix as a transformation,
which we have already talked about it.
When we think about matrices as transformations
they really are functions.
There are functions that are taking one point
in a certain dimensional space
let's say in the coordinate plane, to another point
it transforms a vector to another vector.
For example, let's imagine something that does
a clockwise 90 degree rotation.
And we know how to construct that transformation matrix
which really is a function.
What it does is, in our transformation matrix
we want to say, what do we do with the one zero unit vector?
And what also do we do with the zero one unit vector
when you do that transformation?
Well, if you're doing a 90 degree clockwise turn,
then the one zero unit vector
is going to go right over here.
And so that's going to be turned
into the zero negative one vector.
So I'll write that right there.
And then the zero one vector is going to be turned
into the one zero vector.
So let me write it down.
This is 90 degrees clockwise
and then we can think about
what 90 degree counter-clockwise
would look like
you're going counterclockwise
your original one zero vector right over here
is going to go over here.
It's going to become the zero one vector.
So we will write that right over here.
And then the zero one vector will then become this vector
if you're doing a 90 degree counterclockwise rotation
it's going to become the negative one zero vector
negative one, zero vector.
So in theory these two transformations
should undo each other.
If I do a transformation
that first gets 90 degrees clockwise,
and then I apply a transformation
that's 90 degrees counter-clockwise
I should get back to where we began.
Now let's see what happens
when we compose these two transformations
and we know how to do that.
We've already talked about it.
We essentially multiply these two matrices.
If you were to multiply zero, negative one,
one, zero times zero, negative one, one, zero.
What do we get?
Well, let's see these, this top left
this is composing two, two
by two matrices is equivalent to multiplying them
we've seen that in other videos.
And so first we will look at this row and this column
and that's going to be zero times zero plus one times one.
So that is going to be one.
They're going to look at this row and this column.
So zero times negative one
plus one times zero is just going to be zero.
And then we're going to multiply
this row times each of those columns.
So negative one times zero is zero
plus zero times one is zero
and then negative one times negative one is one
plus zero times zero is one.
And look what happened when we took the composition of
these two matrices that should undo each other
we see that it does.
It turns into the identity transformation
or the identity matrix.
We know that this matrix right over here as a transformation
it's just going to map everything onto themselves.
Now, this is really interesting because if we view
these two by two transformation matrices as functions,
we've just shown that if we call this
say our first function then can call this it's inverse.
And actually we use that same language
when we talk about matrices.
If we call this as being equal to A
we would call this as being equal to A inverse.
So if I were to take matrix A
and I were to multiply that times its inverse
I should get the identity matrix, which is right over here.
And here I'm speaking in generalities
I'm not even just talking about the two by two case.
That should be the three by three case
the four by four case so on and so forth.
And we also know, that I could have defined this bottom one
as A and the top one as A inverse.
And so the other way should be true as well.
A inverse times A should also be equivalent
to the identity matrix.
And so that's completely analogous to what we saw
in these function examples
between a function and its inverse
because the other day, as we said
an end by end matrix can be viewed
as a transformation can be viewed as a function.
And we also see that it has analogs to just how
we think about multiplication.
'Cause here we could do this multiplication
as a composition of transformations
but we also can just view this as matrix multiplication.
And so if we take a matrix
and we multiply it by its inverse,
that's analogous to taking a number
and multiplying by its reciprocal
and we get the equivalent of
what in the number world would just be one,
but in the matrix world is the identity matrix.
'Cause the identity matrix has this nice property
that if I were to take the identity matrix
and I were to multiply at times any matrix
you're gonna get the original matrix again
which is what we saw at least within the analog that we saw
in the regular number world.