How does this equation have infinitely many solutions?? #math #complexnumbers #youtubemath

I know you're not going to believe me at first, but this equation can actually have infinitely many solutions. Now, I can feel the distrust brewing. So, just give me a couple of minutes and I'll explain. You see, usually when you look at an equation like this, you're thinking about it in a particular context. Maybe you're looking at solutions in the integers or solutions that are real numbers complex numbers. Either way, in these three cases, the only solutions are -1 and one. So like for real, what am I actually talking about here then? Well, when mathematicians look at an equation like this, they ask what is the algebraic context and framework we're actually working in. Here we have a variable that we don't know and a multiplication that's taking place and then we want this multiplication to equal the multiplicative identity in the framework we're working in. Well, what in the world does that even mean? Well, generally in different algebraic frameworks, the number one is what's called a multiplicative identity. That means if we take any number or object in our system, multiplying it by this number one in the system will not change the number we started with. So, can we think of an algebraic framework where this equation will have infinitely many solutions? Well, here is an example of one. the 2x two matrices with real entries. Before we investigate why, we need to understand what all these objects mean in this algebraic land. So first of all, the multiplication in this system is matrix multiplication. So here we're taking a matrix X and multiplying it by itself. Moreover, the identity here in this algebraic system is the identity matrix. So we're asking for matrices X whose square is the identity. Now, here is a relatively simple matrix that satisfies the condition. If we let M be this matrix right over here, then squaring it because it's diagonal can be achieved by just squaring the diagonal entries and we'll get the identity matrix there. So, we at least have one solution to this equation. But now, there's actually a very clever way to generate infinitely many solutions. What we'll do is start with any invertible matrix P and then we'll take this matrix M and do what's called conjugation by P which is multiplying by P on the left and P inverse on the right. So what happens to this matrix P MP inverse when we square it? We'll get P MP inverse times itself. But we notice the internal P inverse and P actually multiplied to the identity. So we're left with P M^ 2 P inverse. But M is this matrix here whose square is the identity. So this gives us P * P inverse which is the identity itself. So starting with this matrix M, we can generate all of these different matrices by ranging over all invertible matrices P. And this gives us infinitely many solutions to this equation.