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## Introduction to matrix inverses

# Inverse matrix introduction

## Video transcript

- [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.