 So let's see if we can prove some theorems about inverse matrices. So today's lesson in etymology is about proof. You didn't think this would be all math, did you? So an older meaning of the word proof is a test. And this is the sense where we talk about proving grounds, where we test munitions for reliability, or we talk about alcohol proof, where the potency of a beverage is tested, or we might even use the phrase proof of concept to show that an idea actually has some merit. And what's important to understand here is that in all of these cases, the concept of proof implies the possibility of failure. A munition might not explode the way it's supposed to. An alcoholic beverage might not be very strong. And the proof of concept may show that the concept is worthless. So let's see if we can prove, in the sense of test out, some of the properties of the inverse. So let's try to prove, or possibly disprove, that P inverse inverse is equal to P itself. So for problems like this, we'll introduce a very serious and important mathematical concept, the duck proof. What's a duck proof? A duck proof is fairly simple. If it walks like a duck, swims like a duck, and quacks like a duck, it's a duck. In this particular case, we want to say that the inverse of P inverse is P, and so we might want to see whether or not these two things act in the same way. And if they act in the same way, there's no reason to say that they are not equal. Now the problem with a duck proof is that we have to know how ducks walk, swim, and quack. So let's ask ourselves this question. Self, what is the property of the inverse? Well, if I ignore what's inside the parentheses, then what I have is the inverse of something. And my definition of the inverse says that if I multiply this inverse of something by whatever it was the inverse of, I should get the identity matrix. And so, putting back our insides, we know that the inverse of P inverse is the inverse of P inverse. And when I multiply the two together, I should get the identity matrix. And that's a very useful property. On the other hand, we know that P times P inverse will be the identity matrix as well. And now let's compare our two statements. The matrix P has the property that if I multiply it on the right by P inverse, I get the identity. Well, it turns out that the matrix, the inverse of P inverse, has the same property. If I multiply this matrix on the right by P inverse, I'll also get the identity matrix. Well, here we have a matrix, and here we have another matrix that walks, swims, and quacks just like it. And so, we are reasonable in concluding that P is the same as the inverse of P inverse. Let's try to prove another property. Suppose we know that P times P inverse gives us the identity matrix. We'd like to prove, or possibly disprove, that P inverse times P also gives us the identity matrix. Now, it's possible that the product might not exist, so we do have to require that the product exists. So what can we do? Well, a good starting point. In any type of problem where we're trying to prove an if-then statement, we can always start with the if claim. So this starts with if P times P inverse equals the identity, and so we get to assume that P times P inverse is equal to the identity. Now, we'd like to say something about P inverse times P. So what we'd want to be able to do is somehow get P inverse times P in the equation, and so I can do that by multiplying both sides on the right by P. And so on the left-hand side, I have P times P inverse times P. On the right-hand side, I'll have the identity matrix times P, which will just give me P back again. And let's use a duck proof. So if I regroup, I notice that over here on the left-hand side, I have the matrix P times something gives me the matrix P. But we also know that P times the identity matrix will give us P. And so because this thing walks like a duck, swims like a duck, and quacks like a duck, it's probably a duck. And so I can conclude that P inverse P and the identity matrix must be the same, and so I, the identity matrix, is equal to P inverse P. Well, that was a lot of fun. Let's see if we can do it again. So let's try to prove or disprove that the inverse of PQ is P inverse times Q inverse. Now, if we go back to our etymology lesson, remember that proof has this concept of test, and test has the possibility of failure. But part of the reason that you run munitions through a proving ground or test the proof of alcohol or run a proof of concept is that if it does fail, it gives you an opportunity to fix what was wrong. And so let's go beyond proving or disproving this statement. If it turns out that it's incorrect, let's find the correct expression for the right-hand side. So let's test it. PQ inverse is the matrix when if I multiply it by PQ, I should get the identity matrix. Well, if PQ inverse is the same as P inverse Q inverse, I should be able to multiply P inverse Q inverse on the right by PQ and get the identity matrix. And so paper is cheap. Let's go ahead and set down the equation, although let's not commit ourselves to saying that it is equal, and so we'll write a question mark above that equal symbol. And we stare at this equation and we realize that we have a problem. We know that P inverse times P and Q inverse times Q are both equal to the identity matrix. However, in general, matrix multiplication is not commutative, and so we have no way of bringing P inverse and P together. Likewise, we have no way of bringing Q inverse and Q together. So there's no way to bring these factors together and make them equal to the identity. So we're stuck here. We can't actually go any further. We can neither confirm nor deny that the statement is true. However, because the proof requires that we test the statement, we can't say that it's true, so we must conclude that it's got to be false. So let's see if we can try to fix it. So we want to find the inverse of the matrix PQ, and so we need to find a product that gives us the identity matrix. So again, we'll start out with our matrix PQ and we want to multiply it by something that will, after all the dust settles, give us the identity matrix. So the only thing I really know at this point is that if I multiply Q inverse by Q, I get the identity matrix. We also know, because we just proved it, that if we multiply Q by Q inverse, we also get the identity matrix. So let's put down a Q inverse and see what happens. And so this Q times Q inverse gives us the identity matrix, and so now I have the product P times the identity matrix, which is just P itself. Well, I'm trying to build a bridge to the identity matrix here. I don't have the identity matrix yet. So what can I do? Well, I also know that P times P inverse will be the identity matrix. So if I supply a factor of P inverse here and then supply it to the other points in the bridge, then I do get, as my final product, the identity matrix. And so here, because PQ times Q inverse P inverse gives me the identity matrix, then it follows that the inverse of the matrix PQ is Q inverse P inverse.