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## Precalculus

# Invertible matrices and determinants

An invertible matrix is a matrix that has an inverse. In this video, we investigate the relationship between a matrix's determinant, and whether that matrix is invertible. Created by Sal Khan.

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## Video transcript

- [Instructor] So let's
dig a little bit more into matrices and their inverses. And in particular, I'm
gonna explore the situations in which there might not
be an inverse for a matrix. So just as a review, we think
about if we have some matrix, A, is there some other matrix
which we could call A inverse that when we take the composition of them, so if we viewed them
each as transformations, we would end up with the
identity transformation. Or if we take the product of the two, you get the identity matrix. And we would also think about it, well, if A inverse undoes A, then
A should undo A inverse to also get the identity matrix. And so another way to
think about it, if I take some type of region in
the coordinate plane, so this is my x axis, this is my y axis. And so let's say my original
region looks something like this right over here. And I apply the transformation
A, and I get something that looks like this, just
making up some things. So if I apply the transformation A, it takes me from that
region to that region. Then we also have a sense
that, okay, A inverse, if you transformed this purple thing, it should take you back
to where you began. Because if you start with
this little blue thing and if you have the composition,
well, then that should just be transforming it with
the identity transformation, so you should just get back to
this little blue thing here. Now this might start
triggering some thoughts about determinants,
because you might remember that the determinant of a matrix tells us how much a region's
area will be scaled by. In particular, let's say that matrix A takes a region that has
an area of, I don't know, let's call this area b and let's take, let's say it takes that area to 5 times b. So the area here is 5b. Well, we know that that scaling of 5 you can determine from the
determinant of matrix A. That would tell us that the absolute value of the determinant of matrix
A is going to be equal to 5. But what does that tell us
about the absolute value of the determinant of A inverse then? Well, if A is scaling up by
5, it scales areas up by 5, then A inverse must be scaling areas down by 5. So the absolute value of the determinant of A inverse should be 1 over 5. And so now we have a general property. I just happened to use
the number five here, but generally speaking, the absolute value of the determinant of matrix
A, if it has an inverse, should be equal to 1
over the absolute value of the determinant of A inverse. And we can of course write
that the other way around. The absolute value of the
determinant of A inverse should be equal to 1 over,
or the reciprocal of, the absolute value of
the determinant of A. The sum comes straight
out of this property that the absolute value of
the determinant tells you how much you scale an area by. Well knowing that both of these
statements need to be true for any matrix A that has an inverse, it gives us a clue as to at least one way to rule out matrices that
might not have inverses. If I were to tell you that the determinant of matrix A is zero, will
that have an inverse? Well, it can't because if
this quantity right over here is zero, or this quantity
right over here is zero, that would mean that the
absolute value of the determinant of the inverse of the matrix
needs to be one over zero, which is undefined. And so we have an
interesting conclusion here. If the determinant of a
matrix is equal to zero there is not going to be an inverse, because let's say that there
was some transformation that determinant was
zero, instead of something that's taking up two-dimensional
area to something else that takes two-dimensional
area, it would transform something that takes
up two dimensional area to something that takes no area. So maybe a curve like that, that takes up no area
or a line or a point. And if you transform to say, a line, how do you transform back? You'd have to scale up the area infinitely in order for it to take up
some two dimensional space. So big takeaway. We've just said, if the
determinant of a matrix is equal to zero, you're not
going to find an inverse. And it actually turns out the
case that any other matrix, you can find an inverse, but
I'm not going to prove that just yet, but hopefully you feel good about this principle right over here.