Let's see if i understand. You wish to find the minimum polynomial without effort?
I don't think there is a royal path here. You have to find (A-6I)^2, (A-6I)^3,... until you get a zero.
If I am given an nxn matrix A and the characteristic polynomial of A is (X-6)^n for A. Is there anyway to work out which of the factors is the minimum polynomial for A by DIRECTLY raising the matrix (A-XIn) to the various powers of the factors (rather than calculating the powers the long way!).