[MUSIC] Welcome back. In this lecture I want to derive an important identity which is called Cassini's identity. The easiest way to drive this identity is making use of the Fibonacci Q matrix, which we talked about in the last lecture. Let's recall what the Q matrix is. Q matrix was this 2x2 matrix that had elements one, one in the first row and one zero in the second row, right? And the interesting derivation we did of the Q matrix was seeing what it looks when you raise it to the nth power. This was the Fibonacci number to the n plus one. The n plus one Fibonacci number. The n Fibonacci number. The n Fibonacci number and the n minus one Fibonacci number, okay? So we had these two results. One is just a definition of the Q matrix, and the other is the result that the Q matrix raised to the nth power is related to these Fibonacci numbers. Okay. So from matrix algebra we have something called the determinant. Remember that the determinant of a matrix. So here, the matrix let's say is q. The determinant of this matrix q for two by two matrixes is very simple right? You take this one and multiply it against zero, and you subtract this one, multiply it against one. So it's the product of the diagonal elements minus the product of the off-diagonal elements. So it's 0- 1. So the determinant of q is -1. Okay. Then there's another result and determinants. So that's a general result. So if you have two matrices A and B okay, and you want to know what the determinant of A times B is. So A and B are N by N matrices here. We only need o worry about two by two matrices. But this is a general theorem that the determinant of the product of two matrices is the product of their determinants. So this is determinant of A times determinant of B, okay? So if you've had linear algebra, you've had some matrix algebra, and you learned determinants. Then this will be familiar to you. Even though you may not remember it exactly. You should know what I'm talking about. If you haven't had matrix algebra, then you can review the basics In the notes for two by two matrices. That's all we need here. Okay? So don't worry. But let's say we know these results. So how can we apply these to the Fibonacci Q matrix to obtain an interesting identity? Well let's look at this. If we look at the determinant of the Q matrix raised the nth power. We can use this previous result as writing this as the determinant Of Q times Q to the N minus one power right, and then split it up. So this is determinant of Q times determinant Of Q to the n minus one. And you see you can keep doing this. You can keep doing this. So you can pull out another Q and get Q to the n minus two. You can pull out another Q and get Q to the n minus three, etc. So that this thing will go all the way down to determinant of Q raised to the Nth power, okay? So the determinant of the matrix Q to the n is equal to the determinant of the matrix Q raised to the nth power, okay? That's the result that we're going to use on the Fibonacci Q matrix. So let's see that, right? So remember, we have our Q, which is one, one, one, zero. And we have our Q to the n, which are obvious Fibonacci numbers. Fn plus one, Fn, Fn, Fn minus n. Right? And the determinant of Q to the n, right. Determinant of Q to the nth power, is F n plus 1, times F n minus 1, right? We multiply the diagonal elements, minus F n squared. We multiply the off diagonal element. So the determinant of this Q to the n is equal to the determinant of Q raised to the nth power right? The determinant of Q is minus one. So this is equal to minus one raised to the nth power, okay? So Fn plus one times F n minus one minus F n squared is minus one to the nth power. That's called Cassini's Identity, okay? Cassini's Identity. So let's see how this actually works. Let's try some numbers. Let's write down some Fibonacci numbers, one, one, two, three, five, eight, 13. That's enough. Let's look at one of these, right. Let's start with three. So Fn+1 would say five, Fn-1 is two, right? So five times two minus three squared equals ten minus nine, right? Ten minus nine is one. Okay, that corresponds to one, two, three, four. The fourth Fibonacci number. So n = 4. So minus one to the fourth is one, right. So let's look at the next one. So n = 5. So we've got 8 Fn + 1 times 3Fn-1 right, minus five squared. What is that? 24 minus 25 right? So that's minus one. Very nice. So this is Cassini's Identity. It tells you that the product of two Fibonacci numbers separated by one minus the square of the one in the middle is going to be the plus or minus one okay. So Cassini's identity will provide for us a very amusing sort of mathematical trick, which I'll show you in the next lecture, okay? See you then.
