 In this video, we'll provide the solution to question number 12 from the practice midterm exam number 2 for Math 2270. We're given a 3 by 3 matrix A, which you can see on the screen. We're asked to compute A inverse and we have to show all of our steps. Now, the algorithm we know to compute the inverse of a matrix is you're going to take that matrix and augment it with the appropriately sized identity matrix. That'll be the 3 by 3 identity in this situation. So, in the first position, we're going to just put the matrix A, 1, 5, negative 3, 1, 6, negative 3, and then negative 1, negative 5, and 4. And then we augment this with the identity 1, 0, 0, 0, 1, 0, 0, 0, 1. And then we have to row reduce this thing. When in doubt, row reduce. So, our first pivot position would be the 1, 1 spot. That's already a 1, which is really grateful. I need to get rid of the 1 below it. So, I'm going to take row 2 minus row 1. So, we're going to get a minus 1, minus 5, plus 3, minus 1. I'm going to ignore 1's add zeros in it. And then to get rid of the negative 1 on the third row, we're going to take the row 3 plus row 1. So, we're going to get plus 1, plus 5, minus 3, and plus 1. And so, performing this row operation, we don't do anything to the first row. So, just copy it down. Negative 5, negative 3, 1, 0, 0. The next one, we're going to get 0, 1, 0, negative 1, 1, 0. And for the third row, we get 0. We're going to get negative 5 plus 5, which is actually a 0. We're going to get 4 minus 3, which is 1. And then we get 1, 0, 1. So, the next step here would be, well, we normally would come to this pivot position right here, but we can see there's already a 0 under it, so that's actually great. And we already have a 1 there. So, I'm actually going to move to the pivot position in my third stage right there. Notice that our matrix right here is already initial on form. That's great. We got that pretty quickly. So, now we're going to start with the backwards face. We want to get rid of the negative 3 that's above. So, we're going to take row 1 and add to it 3 times row 3. So, we get a 3, 3, and 3. And that's the only thing I need to do in that stage. So, we're going to record this down. The first row becomes 1, 5, 0, and then you're going to get 4, 0, 3. I didn't do anything to the second row, so we'll just copy it down. 0, 1, 0, negative 1, 1, 0. And then for the third row, 0, 0, 1, 1, 0, 1. Again, we didn't do anything to that one. So, now we move our pivot position back to the 2, 2 spot. We got rid of that 5 that's right there. So, we're going to take row 1 minus from it 5 times row 2. So, we're going to get a minus 5. We're going to get a plus 5 and a minus 5 right there. And so, recording what we have, we're then going to get on the left side of our matrix and get 1, 0, 0, 0, 1, 0, 0, 0, 1. For which that's the identity which we basically are done. Looking on the right hand side, we're going to get 9, negative 5, 3, negative 1, 1, 0, and 1, 0, 1. Like so. Now, don't stop here because we haven't actually identified what the matrix is yet. We need to actually be specific. A inverse is equal to the matrix which we see right here. Again, actually write this down. Don't just assume that I know that you know this one. You got to be explicit here. So, A inverse equals 9, negative 5, 3, negative 1, 1, 0, and 1, 0, 1. So, that's what you would record as your answer on this one. Now, it's important also when you do things like this, I would actually recommend checking your answer if all possible, right? So, if we were to check our answer, we take A times A inverse. So, remember the matrix A was given, right? You can't see on the screen right now. Let me write down. You get 1, 5, negative 3. You're going to get 1, 6, negative 3. And then negative 1, negative 5, and 4, like so. And we didn't multiply that by A inverse which was on the screen a moment ago. But just recording what we have, we have 9, negative 5, 3, negative 1, 1, 0, and 1, 0, 1. So, multiply these things out and when you go through all the possible combinations, you're going to see that we actually get the identity, right? And it's good to check these things. So, with the first box, when you take the first row, first column, you get 9, minus 5, minus 3. So, notice right there. I probably shouldn't do that yet. But notice that's going to equal a 1. When you do the next spot, the first row, second column, you're going to get negative 5 plus 5. And for the first row, third column, you end up with a 3, minus 3. So, in both of those situations, you get 0, 0. Exactly what you'd expect. So far, so good. If we do this with the second column, you're going to get 9, minus 6, minus 3. That's a 0. Second column, second row, second column, you're going to get negative 5 plus 6 plus 0. That's a 1, of course. And then for the next bit right there, the second row, third column, you end up with a 3, minus 3 again. And so, that gives us a 0. And so, lastly, if you do the third row, you're going to get negative 9 plus 5 plus 4. That's obviously a 0. And then for the last one, you're going to, excuse me, the second to last one, third row, second column, you're going to get 5, minus 5. And then that's a 0. And then the last one, you can probably guess what's going to happen here. We're going to end up with negative 3 plus 4, which is equal to a 0. So, we do in fact have the inverse matrix like we were looking for.