Problem Statement:

Determine for .

First compute the eigenvalues

or (repeated roots). Since the roots are identical, the confluent form of Sylvester's theorem must be used. Let's first compute the term . Recalling, the definition of an adjoint matrix, one has

Now since (for the case where all the roots are equal), we have

Taking the first derivative of this expression with respect to , gives

Evaluating this expansion at and recognizing that the sum in eqn. (2.44) only has a single term, gives