A matrix M has eigenvectors (3,1,0) (2,8,2) (1,1,6) with corresponding eigenvalues 1, 6, 2 respectively. Write an invertible matrix P and diagonal matrix D such that M=PD(P^-1), hence calculate M^5.

Without even knowing M, the candidate can calculate M^5. This will follow from the fact that P is the matrix consisting of the eigenvectors of M as columns, and D will have the eigenvalues (in matching columns to their corresponding eigenvectors) down the lead diagonal. The candidate will have to do some computation to determine P^-1, but this is standard in A-level and will serve as good practice.Then we see that M^5 = (PD(P^-1))^5 = P(D^5)(P^-1), the essence behind this being that D^5 is very simple to calculate since D is diagonal.Again this final stage requires some computation, but getting comfortable with this serves as a great means to reduce the pressure of time in the exam.