 Welcome back to our lecture series, linear algebra done openly. As usual, I'll be your professor today, Dr. Andrew Misseldine. In section 3.3, we wanna answer the following question. When can we divide by a matrix? I mean, doesn't even make sense to talk about matrix division. In the previous sections of this chapter, we've introduced matrix addition, subtraction, matrix multiplication even, but can we undo multiplication? With division, well, we've seen some situation where that's not gonna happen, and so it really does depend on the matrix. Some matrices we can and some we cannot. This leads to our definition. An in-by-in matrix is called non-singular, or sometimes it's called invertible. Invertible, the name kinda makes sense right here. We say that a matrix is non-singular if there exists some other matrix, say B, such that A times B is equal to B times A, so these matrices do have to commute. I'm not saying that A commutes with every matrix, but it will commute with this matrix B because in particular, A times B and B times A will equal the in-by-in identity matrix, and we call B an inverse of the matrix A. If a matrix is non-singular, we call it singular. It's kind of like a double negative there. Now, a matrix being singular, the term here kinda means that if it's singular, there's something kind of wrong with it. Like it contains a singularity. Like in the cosmos, singularity is often represented black holes and things. There's like a hole. There's something wrong with the matrix. So singular is not a good thing in this context. So if you're non-singular, that means you don't have that singularity that kinda leads to division by zero and thus you can't divide by the matrix. So a matrix is non-singular. It's invertible if it has an inverse. And so I wanna give you an example of such a thing. Consider the matrices A, which is two by two, it's two, three, three, five. And C is also a two by two matrix, five, negative three, negative three, two. Now I wanna mention that we defined non-singular matrices only for square matrices. If you're not square, then you can't be non-singular. So a non-square matrix is automatically gonna be singular. Now in this situation, we have two square matrices. What happens when we multiply them together? Like if we take A times C, notice this will look like two, three, three, five times five, negative three, negative three, and two. If we go about with the matrix multiplication, so we're gonna take the first row times the first column. We end up with 10 minus nine. Then we're gonna do the first row times the second column here. We're going to get negative six plus six. And then if you do the second row times the first column, we end up with 15 minus 15. And lastly, the second row times the second column, we end up with negative six, excuse me, negative nine, three times three plus 10. And when you simplify that, you'll notice you end up with 1001, which is the two by two identity. So this shows us that A times C is equal to the identity. Now to be an inverse, we need to the other way around, right, C times A. If we take five, negative three, negative three, and two, and you multiply that by two, three, three, five, first row first column, you're gonna get 10 minus nine, which notice that's a one. The first row second column, you're gonna get 15 minus 15, which is a zero. The second row first column, you get negative six plus six, which is a zero. And then lastly, second row second column, you're gonna get negative nine plus 10 right here, which is the two by two identity, once again. So we can see in this situation that A is in fact non-singular. It's a non-singular matrix. And it has an inverse, which is this matrix C right here. So we found example of a non-singular matrix. I wanna mention that when it comes to non-singular matrices, we first defined that it had an inverse, but it turns out we should be using the article the, that is to say that if a matrix is invertible, the inverse is actually unique. There's only one matrix that'll be the inverse. And so what I want you to do is kind of consider the following, let's take a matrix A, and let's say that B and C are both inverses of A. So I'm gonna times A by B and by C. Now, when you have a triple product right here, we have to do one of the products first. So we could think of B times A, C, or we could think of this as B times A, C. Now, as the matrix multiplication is associative, it doesn't matter which one we do, we'll get the same thing. Now, if A, if C is an inverse of A, that means A times C is equal to the identity. So you get B times the identity, IN, and the identity matrix times any matrix will just be that of the matrix. So in one regard, I think this product should be B. But on the other hand, if you look at B times A, since B is also an inverse of A, B times A will equal the identity, and the identity times any matrix would be C. I mean, times any matrix would be that matrix, which in this case is C. And so then you see that the matrices B and C actually have to be one and the same thing. So in fact, inverses of a matrix are unique. And so we refer to the inverse, we refer to the inverse of A as, we're gonna write this A to the negative one power there. So A inverse, and this is the idea that, like when we talk about real numbers, if you take something to the negative one power, that gives you its multiplicative reciprocal. It's a multiplicative inverse, so we want the same thing for matrix multiplication here. And so we see that when you multiply a matrix by its inverse, this will equal A inverse A and this will equal the identity matrix when a matrix is non-singular. Now of course, if A is a singular matrix, A inverse doesn't make any sense. A inverse doesn't exist if your matrix is singular, but it exists exactly when the matrix is non-singular.