Determining Improvements: Cassini's Identity

Three squares is pretty good, but we can do even better!

Let's look at Mn=(F(n+1)F(n)F(n)F(n−1))M^n = \begin{pmatrix} F(n+1) & F(n) \\ F(n) & F(n-1) \end{pmatrix} again. There's a tool from linear algebra we can use here called the determinant, notated as det⁡M\det M. I'll skip the geometric intuition behind the determinant because it's not especially relevant here, but suffice it to say that the determinant of a matrix is a number that represents how much the matrix scales its inputs.1 (The identity matrix has determinant 1, for example, because it preserves areas.)

Importantly, determinants play nice with multiplication: det⁡AB=det⁡Adet⁡B\det AB = \det A \det B. For a 2×22 \times 2 matrix (abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix}, the determinant is ad−bcad - bc.

Let's use this tool to discover a new equation that can help us out.

  1. For more, consult any resource on linear algebra. Here's a good one.↩

Determining Improvements: Cassini's Identity

Three squares is pretty good, but we can do even better!

Let's look at Mn=(F(n+1)F(n)F(n)F(n−1))M^n = \begin{pmatrix} F(n+1) & F(n) \\ F(n) & F(n-1) \end{pmatrix} again. There's a tool from linear algebra we can use here called the determinant, notated as det⁡M\det M. I'll skip the geometric intuition behind the determinant because it's not especially relevant here, but suffice it to say that the determinant of a matrix is a number that represents how much the matrix scales its inputs.1 (The identity matrix has determinant 1, for example, because it preserves areas.)

Importantly, determinants play nice with multiplication: det⁡AB=det⁡Adet⁡B\det AB = \det A \det B. For a 2×22 \times 2 matrix (abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix}, the determinant is ad−bcad - bc.

Let's use this tool to discover a new equation that can help us out.

  1. For more, consult any resource on linear algebra. Here's a good one.↩