 In this video, we will present the solution to question number eight from the practice midterm exam for math 2270. We're asked to solve the matrix equation A minus AX all raised to the inverse is equal to X inverse plus B. Now, while we're solving this equation, we may assume that all the matrices are in by in and we also may assume that every matrix in play is non-singular even if that is a ridiculous assumption. That is, we can take inverses without any concern here while we should be concerned about multiplication commuting because matrix multiplication general does not commute. Another thing to also worry about here is that a common mistake is that the inverse does not distribute across a sum. We don't have a property that does that. This would be like saying like, if you have one over two minus three, this is not the same thing as one half minus one third. This would be the matrix analog of doing such a thing. So what I actually wanna do is I wanna start off by taking the inverse of both sides. I'm gonna take the inverse of the left and I'm gonna take the inverse of the right. What's good for the goose is good for the gander. You have to make sure you do it to both sides. Now, if you take the inverse of the inverse, that's just the original function. So the left-hand side turns into A minus AX. We're trying to solve for X right here. On the right-hand side, we're gonna have X inverse B inverse. So while we cannot distribute inverses or powers in general across the sum or difference, that's not the case for products. If you take the inverse of a product, we can do something with that. This turns into B inverse X. We'll be inverse inverse, but the double inverse is the matrix again. So notice how things got twisted around. This is a consequence of the shoe sock principle. So we wanna solve this matrix equation for X. So I would then add AX to both sides. Add AX to both sides. This then gives us that A is equal to, we'll take AX plus B inverse X. I can then factor out the X on the right-hand side. So we're gonna get A plus B inverse times X is equal to A. And then basically to solve for A here, we're gonna multiply both sides on the left by A plus B inverse inverse. So this would then give us that X equals, make sure you stay on the left here, A plus B inverse inverse times A. And so this gives you one solution you could come up with. Now I want you to be aware that if you solve this equation in a slightly different manner, you might actually get what looks like a different answer. Again, there's a lot of possibilities on what the solution could look like. Again, this is one of them. An alternative you could do is this is also equal to say the identity plus B A inverse times B A like so. So this is just a, like I said, there might be more than one way. So if you're trying to work this one on your own and you didn't get something that matched up with what you see on the screen right here, that doesn't necessarily mean you did something wrong. You just might wanna double check that these matrices are in fact equivalent or the matrix you got was in fact equivalent to that.