Invertible matrices and determinants | Matrices | Precalculus | Khan Academy
So let's dig a little bit more into matrices and their inverses, and in particular, I'm going to 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 view 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.
We would also think about it, well, if A inverse undoes A, then A should undo A inverse to also get the identity matrix. 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, 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 five, it scales areas up by five, then A inverse must be scaling areas down by five.
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 happen to use the number 5 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 could 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. This 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 1 over 0, which is undefined.
And so we have an interesting conclusion here. If the determinant of a matrix is equal to 0, there is not going to be an inverse. Because let's say that there was some transformation that determined it 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 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.