Several days ago, I placed a question on MathLinks asking the relation between and . The point is how to evaluate
Interestingly, K.M. showed me a new way to attack such a problem but slightly different from the original one. He proved
Let us discuss the proof of this modified problem.
The determinant we are trying to compute is
which is the characteristic polynomial of evaluated at .
Now, is certainly diagonalizable (which doesn’t even matter, but it makes it easier to think about), and we know its eigenvalues. Why do we know its eigenvalues? Because is a matrix of rank 1, hence nullity , hence of its eigenvalues are zero. What is the other eigenvalue? It’s the same as the sum of the eigenvalues, which is the trace of , which is . Put that information together, and we have that the characteristic polynomial of is
Substitute for to get the result quoted.