Determinant of the Wronskian
The Wronksian of functions is the matrix determinant . Its derivative is the matrix determinant (that is, the previous matrix with a different bottom row). It’s an interesting exercise to prove this, so let’s do that.We proceed by our old friend, induction. For (or 0), the case is obvious. Let it be true through . Expand by the bottom row:
We take the derivative, applying our induction assumption, obtaining
But the bracketed part is just , which is zero as a matrix with repeated rows is singular. We are done.