 Now, this is where we left off. We had this example. We had x prime equals 8 times x, x prime equals 8 times x. Now, we need to solve for x. x would be, let's just make sure, what is that saying to us? That it's going to be x prime, y prime equals, well it is this going to be this one times x plus 3 times y and 5 times x plus 3 times y. That's what we're saying to ourselves. So we have this linear set. x prime equals x plus 3y and y prime equals 5x plus 3y. That's what we're trying to solve. And now we need to get x. Now remember this from in the algebra, I can multiply both sides by, I can multiply both sides by a inverse, so that's going to lead me, a inverse times a times x and I've solved for x. The only problem is, well, I can definitely get the inverse of a, but what is the, what is x prime? I'm not given x prime. I'm not given x prime explicitly. I'm giving it, I'm given it in terms of x's and y's. I've been giving it in terms of x's and y's. So I've got to think of another way to solve this problem here. Okay, so let's just remind ourselves of something. These were the two solutions that we, that we saw in the previous example. But look at it. It's e to the power something times t, e to the power something times t. And I have these. Let's just call this x sub i, which i would be 1, 2, many of the words. Yes, it's 2, so I would just have k1, k sub 1 and k sub 2, k sub 1, k sub 2, k sub 1, k sub 2, e to the power something specific times t. Or if I just write it generally, I have this x equals this vector k, column vector, times e to the power lambda t. So what would x prime be? Well, that is just going to be, I'll bring the lambda forward, lambda k, e to the power lambda t. That would be x prime. But remember, we said that that x prime equals 8 times x. 8 times x. So what is x prime? Although we have x prime, it's k, lambda times k, e to the power lambda t. And that equals a times x. And there's that. That's a times x is there. That's k, e to the power lambda t. I can take lambda t after both sides. And this is what I'm left with. I'm left with lambda k equals 8k. I can just bring this to the other side. Let's just have that. We have 8k minus lambda k equals 0. Watch out now. Lambda, there, this is the scalar. It's not a vector in there. That's a scalar. So that is a scalar times a vector. This is a vector times a vector. Let me say scalar times a matrix and a matrix times a matrix. I've got to change this before I can take k out. It's very easy to do. I'm going to have 8 minus a lambda i k equals 0. So I've multiplied the scalar by the unit matrix. And if I do that, it just remains what it is. Now it is a matrix minus a matrix. I can't have a matrix minus a scalar. Okay, so I've had to have put that in. Now the only time remember when you can have a matrix times a matrix equals 0 is the determinant of this one. The determinant of this A minus lambda i has got to equal 0. A minus lambda i has got to equal 0. So what do we have here? So A is this 1, 3, 5, 3 minus lambda times pi. So remember what is lambda and then i is going to be 1, 0, 0, 1. If I multiply that, that's lambda 0, 0, lambda. So minus lambda 0, 0, lambda. So I'm left with 1 minus lambda 3, 5, 3 minus lambda. And what I need to do now is to get the determinant of that. Now let's wipe the board quickly. C4. Right, let's get the determinant of this. And the determinant of that is going to be 1 minus lambda and 3 minus lambda. And we're going to subtract from that 15. And that's going to equal 0. And that's going to equal 0. In other words, 3, we have minus 3 lambda. We have another minus lambda. And we have a lambda squared, negative 15. That's going to equal 0. So we have lambda squared minus 4 times lambda. And we have negative 12. It seems negative 12 is going to equal 0. And that means lambda. Let's have a lambda minus 6 and a lambda plus 2 is going to equal 0. Is that right? So we'll have a lambda sub 1 equals 2 and a lambda sub 2 equals 6. And those are called the characteristic values or the eigenvalues of the linear set that we're dealing with here. Good.