 In this video, we are going to present the solution to question number 5 from the practice midterm exam number 2 for math 2270. We're given a 3 by 3 matrix A right here, 0, 1, 2, 1, 0, 3, 4, negative 3, 8. And we have the sequence of row operations that converts A into the identity. So we know this is a non-singular matrix. We're supposed to factor A as a product of elementary matrices. And so we notice here as we go from A to this matrix to the next one, to the next one, to the next one, to the next one, to the identity. Each step along the way, there is a single row operation that's performed. And so what I want to first do is recognize what those operations are. To go from A to the first matrix there, though, I've performed an interchange operation. Okay, so you interchange rows one and two. I'm gonna keep track of my pivots right here. So we just went from, we interchange rows one and two. To construct the elementary factorization of A, I'm gonna get a matrix going left to right. It's gonna be the same row operations we have to do the inverse. Now the inverse of the interchange matrix is its own, it's its itself. So the interchange matrix, it'll look like the identity, but we swapped the, we swapped the order of rows one and two right here. So you're gonna get one, or 0, 1, 0, 1, 0, 0, 0, 1. This is an interchange matrix right there to coincide with the operation and play right there. So next, as we go from this matrix to that matrix right there, notice that we're getting rid of the four right there. We took row three, excuse me, and then we subtracted from it four times row one. That's how we went from here to here. So what we did here is we have a minus four, zero, and then a minus 12 right there. We don't actually have to do the details of that. That's what the operation plays here. So the next matrix in the factorization is gonna be a replacement matrix for which we're gonna get, it's a unit lower triangular matrix here. So we get ones on the diagonal, zero's above. And so in the three one position, we're gonna get a positive four. So the thing is we have to take the inverse operation. So if we took minus four row one, we're gonna get a plus four in the matrix right here. So that's, that's how this elementary factorization works. As you, you go in the same order left to right, but you have to do inverse operations. All right, so then look into the next matrix right here. We see that the first column's done. So my pivot position now would come to this one right here. I wanna give her that negative three that's below it. So I would take row three plus three times row two. That's another replacement matrix that we're gonna put here. And again, it's gonna be lower triangular, unit lower triangular. And you're gonna get zeros below the diagonal except in the three two position, in which case I'm gonna get a negative three. I have to do the opposite of the number I see right here. Then we get this matrix right here, great. Your pivot position's gonna be right here. So how do you go to the next matrix? Well, notice the pivot went from a two to a one. So that suggests to me that scaling happened. We took the second, the third row and we divide everything by two, right? So the next matrix in this factorization will be a scaling matrix. So it's gonna, it's gonna be a diagonal matrix. So everything off the diagonal will be zero. Everything on the diagonal will be one except for the row that we changed. We changed row three, so we divided three by two. So we have to put the inverse operation. So we're actually gonna times it by two. So you get one, one, two along the diagonals, okay? So the next thing here is that looking at this pivot, what's the next thing to do? Notice they got rid of the three right here, went from three to zero. So there must have been a replacement operation that happened there. So this would be row one minus three times row three. So our replacement elementary matrix, it'll be unit triangular. But this time it's gonna be upper triangular because we're taking, we're adding to a lower number matrix row there, excuse me. And this is gonna be in the one three spot. So in the one three spot, we're gonna put a positive three because we have a negative three right there. And then we have one more operation to do. Notice that our pivot position as we go from this matrix to the identity, we gotta get rid of this two right here. So how is that accomplished? That's accomplished by taking row two minus two times row three. I'm gonna scoots this over a little bit. So we got this last matrix. This is a replacement matrix, so it's gonna be unit upper triangular. We're gonna get zeros everywhere except for the two three position, the two three position. So we have a negative two coefficient, so we're gonna put a positive two right there. And I'm gonna zoom out a little bit so you can see all of it here. And so now there are six row operations to get from A to the identity, and therefore our factorization will have six elementary matrices, which we had the interchange, replacement, replacement, scaling, replacement, replacement. And in each of these situations, we take this inverse operation. The inverse of interchange is itself, so you don't have to worry about that. For replacements, you have to switch to sign. So we got a positive four, negative three, positive three, positive two. And then for scaling, you just have to take the reciprocal. We times by one-half, so in my factorization, I get a two.