Cassini’s identity for Fibonacci numbers says that . The classic proof of this shows (by induction) that . Since , Cassini’s identity follows.
In this post I’m going to give a different proof involving determinants, but one that does not use powers of the Fibonacci matrix.
Let’s start with the identity matrix, which we’ll call :
To construct , add the second row to the first and then swap the two rows. This gives us
Continue this process of adding the second row to the first and then swapping the two rows. This yields
Since and , the fact that we’re adding rows each time means that .
Since adding a row to another row doesn’t change the determinant, and swapping two rows changes only the sign of the determinant, we have
which is Cassini’s identity.
See also my paper “Fibonacci Identities via the Determinant Sum Property.”